GRAPH A GRAPH B After looking closely at the graphs (or at these full-size images of Graph A and Graph B), think about these questions: • What do you notice?
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Graph A and Graph B (below) originally appeared elsewhere on NYTimes.com.
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Show the data as a bar graph or other type of graph.
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Graph A, above, and Graph B, below, originally appeared elsewhere on NYTimes.com.
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"If you consider that LinkedIn is the "business graph," and Facebook is the "social graph," then Nextdoor is building the platform for the "local graph.
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Wrong. They are about resumes, not motivations Today's professional networks have a "resume graph" or (to be flattering) a "prospect graph," not a "[people's economic] motivation graph" (the graph that plots their give/get instincts over time and in different contexts).
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It shows a graph of sea-surface temperatures, misleadingly implying that it's a graph of global temperatures (a move the scientist who produced the original graph called "very misleading").
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The company offers graph-power business applications like graph-enabled artificial intelligence and machine learning.
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If you ask me, there's not a more aesthetically pleasing graph than the bar graph.
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TIMELINE GRAPH A timeline is a graph of a line with a sequence of dates.
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This is done by recommending content based on a semantic graph and a social graph.
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DIRECTED GRAPH In discrete math, a directed graph shows the relationship between pairs of objects.
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Analyze a Sports Graph: In a recent edition of our What's Going on in This Graph?
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It also includes its own graph database engine called the Cray Graph Engine, which the company claims is ten to 100 times faster than current graph solutions running complex analytics operations.
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In fact, the social graph is almost non-existent in Pokémon Go. Instead, your in-game social graph is an extension of a supplemented version of your real-world social graph.
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If you're interested in exploring this graph further, join this week's What's Going On in This Graph?
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The resulting graph is startling: The graph starts in 1000 BC and goes to the present day.
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Beyond the professional graph and economic graph, the tool with the most potential power is one that enables individuals to actively drive their own workplace learning and career growth through the experience graph.
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WEINER: AND JON, YOU CAN ACTUALLY THINK ABOUT THIS MARRIAGE OF MICROSOFT'S CORPORATE GRAPH AND LINKEDIN'S PROFESSIONAL GRAPH AS A HUGE LEAP FORWARD FOR THE ECONOMIC GRAPH VISION DIGITALLY MAPPING THE GLOBAL ECONOMY.
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This week the hot quiz, at least in my social graph, is the "how millennial are you?" graph.
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The "Explore the Power of the Places Graph" session's description reads: Power your app with the Places Graph.
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EditorsNote: corrects "starting" in Graph 4, and fixes ball 4 to pitch outside the zone in Graph 6.
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" — Morgan (@morganknutson) October 12, 2018 "Google built the knowledge graph, and Facebook swooped in and built the social graph.
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If the links between nodes are bidirectional, we say that the graph is undirected; otherwise, it's a directed graph.
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First I had students talk with a partner about the following: what they noticed about the graph, what they wondered about the graph and what they thought the author was trying to communicate with the graph.
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The database service supports graph models Property Graph and W3C's RDF and their query languages Apache TinkerPop Gremlin and SPARQL.
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" That, in turn, is to be mated with LinkedIn's "professional graph," producing a new thing, "the world's first economic graph.
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The food eaten graph is a frequency bar graph showing the average greenhouse gas impact for 50 grams of protein.
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The commuter delay scarf graph is neither a stacked bar graph nor a time series, but incorporates aspects of each.
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DataStax Enterprise Graph is actually a set of products that includes DataStax Enterprise Server for the graph database component, DataStax OpsCenter for the management piece, DataStax Studio for graph visualization and DataStax Drivers for various language support.
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The most well-known graph database is the Facebook Social Graph, which lets you see connections between yourself and your friends.
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Maybe Mr. Appleseed was such a huge graph lover that he needed four volumes just to do his graph passion justice.
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The food wasted graph is a relative frequency bar graph showing the percentage of food wasted or lost by food category.
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Here's what jumped out at me looking at this graph: The yellow and green dots cluster toward the top of the graph.
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Solomon talks of a "donor graph" — not unlike Facebook's social graph where you can figure out a map of all your connections to people and interests, or LinkedIn's professional graph that tries to create the same picture for your work life.
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When DataStax acquired Aurelius, a graph database startup last year, it was clear it wanted to add graph database functionality to its DataStax Enterprise product, and today it achieved that goal when it announced the release of DataStax Enterprise Graph.
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Steam could add some form of usefulness to this graph by linking highly shared articles about the game to the graph, providing context.
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This week's graph is a directed graph (see Stat Nuggets for full explanation) of this surge in passenger demand during this one week.
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Bellumio says they developed this approach based on the team's collective expertise on deploying large-scale graph databases, machine learning and graph analytics projects.
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In the Thanksgiving passenger graph, the directed graph shows the increase in traffic for Thanksgiving between flight origins and destinations, which are the vertices.
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Then, use these three questions from our "What's Going On in This Graph?" feature to help you interpret the graph: What do you notice?
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Now, compare this to the percentage from this National Youth Risk Behavior Survey graph (PDF, page 11) which predates the New York Times graph.
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The idea of a graph was first popularized by Facebook, which discussed an individual's social graph, the connections they had between friends on the social network.
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Adding in some of these connections makes the graph look like this: The above graph is more complex than the previous, and it provides more information.
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TIME SERIES GRAPH A time series graph displays a series of successive quantitative data points over a time period, which is usually divided into equal intervals.
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The upward-trending line graph usually indicates good news, and graph enthusiasts will often use this emoji to say something in their life is going well.
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For us, that's a particularly interesting challenge right now because we're moving beyond just a follow graph, an account follow graph, to also a topic taxonomy.
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Instead of looking at the title and description to make sense of the graph, try focusing on the axes and the content of the graph itself.
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It's not like the entire graph like your friends and family but it's a graph on what chat you are in, so where are you discussing something.
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The graph below shows where people were tweeting about vote rigging on the day after the election — a noticeable decrease in activity compared to the graph above.
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Graph: Internet Archive Graph: Internet Archive A few years ago, I wrote a piece in which I lamented that the internet was failing the website preservation test.
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All Dreamers were less than 16 years old upon arrival in the U.S. The Dreamers state graph is a bar graph with the 14 states as categories.
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After looking closely at the graph above (or at this full-size image) and possibly your state's graph, think about these three questions: • What do you notice?
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In doing so, Huh found that when he constructed a singularity from a graph, he was suddenly able to use singularity theory to justify properties of the original graph—to explain, for instance, why the coefficients of a polynomial based on the graph would follow a log concave pattern.
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Take this graph of where Americans are getting news, for example: The graph cites the well-known Pew Research Center, along with the date the information was gathered.
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Perhaps finding a smaller five-color graph — or the smallest possible five-color graph — would give researchers further insight into the Hadwiger-Nelson problem, allowing them to prove that exactly five shades (or six, or seven) are enough to color a graph made from all the points of the plane.
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EditorsNote: Corrects punctuation for Shin-Soo Choo in 20th graph, adds Joey Gallo's first name in 03th graph, typo in 20th graph Ronald Guzman hit a three-run homer to cap a seven-run fifth inning as the Texas Rangers defeated the host Seattle Mariners 211-20 on Tuesday night.
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In the Labor-force Participation Rate graph, the variable was labor-force participation percentage and the graph shows how it changes from 1980 - 2017 for seven selected O.E.C.D. countries.
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Stat Nuggets for "Read a Hurricane Map the Right Way" MAP as Graph A map can be a graph when the map shows statistics tied to certain geographic regions.
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Every distance-hereditary graph is a circle graph, as is every permutation graph and every indifference graph. Every outerplanar graph is also a circle graph.; .
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The Kneser graph is the complete graph on vertices. The Kneser graph is the complement of the line graph of the complete graph on vertices. The Kneser graph is the odd graph ; in particular is the Petersen graph.
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In graph theory and graph drawing, a subhamiltonian graph is a subgraph of a planar Hamiltonian graph...
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The Petersen graph is a cubic graph. The complete bipartite graph K_{3,3} is an example of a bicubic graph In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs.
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In graph theory a process graph or P-graph is a directed bipartite graph used in workflow modeling.
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The cubic distance-regular graphs have been completely classified. The 13 distinct cubic distance-regular graphs are K4 (or tetrahedron), K3,3, the Petersen graph, the cube, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the dodecahedron, the Desargues graph, Tutte 12-cage, the Biggs–Smith graph, and the Foster graph.
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In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph..
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The Grötzsch graph shares several properties with the Clebsch graph, a distance-transitive graph with 16 vertices and 40 edges: both the Grötzsch graph and the Clebsch graph are triangle-free and four-chromatic, and neither of them has any six- vertex induced paths. These properties are close to being enough to characterize these graphs: the Grötzsch graph is an induced subgraph of the Clebsch graph, and every triangle-free four-chromatic P_6-free graph is likewise an induced subgraph of the Clebsch graph that in turn contains the Grötzsch graph as an induced subgraph . The Chvátal graph is another small triangle-free 4-chromatic graph. However, unlike the Grötzsch graph and the Clebsch graph, the Chvátal graph has a six-vertex induced path.
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In the mathematical field of graph theory, the bull graph is a planar undirected graph with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendant edges. It has chromatic number 3, chromatic index 3, radius 2, diameter 3 and girth 3. It is also a self-complementary graph, a block graph, a split graph, an interval graph, a claw-free graph, a 1-vertex- connected graph and a 1-edge-connected graph.
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Otherwise, it is called a weakly connected graph if every ordered pair of vertices in the graph is weakly connected. Otherwise it is called a disconnected graph. A k-vertex-connected graph or k-edge-connected graph is a graph in which no set of vertices (respectively, edges) exists that, when removed, disconnects the graph. A k-vertex-connected graph is often called simply a k-connected graph.
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Every distance- hereditary graph is a perfect graph,. more specifically a perfectly orderable graph, pp. 70–71 and 82. and a Meyniel graph.
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Two drawings of the And(4) graph In graph theory, an Andrásfai graph is a triangle-free circulant graph named after Béla Andrásfai.
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Threshold graphs are a special case of cographs, split graphs, and trivially perfect graphs. Every graph that is both a cograph and a split graph is a threshold graph. Every graph that is both a trivially perfect graph and the complementary graph of a trivially perfect graph is a threshold graph. Threshold graphs are also a special case of interval graphs.
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In the mathematical field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle. The triangle graph is also known as the cycle graph C_3 and the complete graph K_3.
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In the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph").
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A cactus graph. The cacti form a subclass of the outerplanar graphs. Every outerplanar graph is a planar graph. Every outerplanar graph is also a subgraph of a series-parallel graph., p. 174.
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The complement graph of every comparability graph is also a string graph. and . See also .
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One definition of an oriented graph is that it is a directed graph in which at most one of and may be edges of the graph. That is, it is a directed graph that can be formed as an orientation of an undirected (simple) graph. Some authors use "oriented graph" to mean the same as "directed graph". Some authors use "oriented graph" to mean any orientation of a given undirected graph or multigraph.
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For each natural number n, the edgeless graph (or empty graph) \overline K_n of order n is the graph with n vertices and zero edges. An edgeless graph is occasionally referred to as a null graph in contexts where the order-zero graph is not permitted. It is a 0-regular graph. The notation \overline K_n arises from the fact that the n-vertex edgeless graph is the complement of the complete graph K_n.
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A distance-hereditary graph. In graph theory, a branch of discrete mathematics, a distance-hereditary graph (also called a completely separable graph). is a graph in which the distances in any connected induced subgraph are the same as they are in the original graph. Thus, any induced subgraph inherits the distances of the larger graph.
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A graph with two nontrivial strong splits (top) and its split decomposition (bottom). The three quotient graphs are a star (left), a prime graph (center), and a complete graph (right). In graph theory, a split of an undirected graph is a cut whose cut-set forms a complete bipartite graph. A graph is prime if it has no splits.
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The result is the Coxeter graph. This construction exhibits the Coxeter graph as an induced subgraph of the Kneser graph . The Coxeter graph may also be constructed from the smaller distance- regular Heawood graph by constructing a vertex for each 6-cycle in the Heawood graph and an edge for each disjoint pair of 6-cycles.. The Coxeter graph may be derived from the Hoffman-Singleton graph. Take any vertex v in the Hoffman- Singleton graph.
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Dodecahedral graph. A 4:2-coloring of this graph does not exist. Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring.
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In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.. Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number s such that each two oriented paths of length s can be mapped to each other by exactly one symmetry of the graph. He showed that s is at most 5, and provided examples of graphs with each possible value of s from 1 to 5.. Semi-symmetric cubic graphs include the Gray graph (the smallest semi-symmetric cubic graph), the Ljubljana graph, and the Tutte 12-cage. The Frucht graph is one of the five smallest cubic graphs without any symmetries: it possesses only a single graph automorphism, the identity automorphism..
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Square grid graph Triangular grid graph A lattice graph, mesh graph, or grid graph, is a graph whose drawing, embedded in some Euclidean space Rn, forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid.
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Therefore, the simplex graph contains within it a subdivision of G itself. The simplex graph of a complete graph is a hypercube graph, and the simplex graph of a cycle graph of length four or more is a gear graph. The simplex graph of the complement graph of a path graph is a Fibonacci cube. The complete subgraphs of G can be given the structure of a median algebra: the median of three cliques A, B, and C is formed by the vertices that belong to a majority of the three cliques.
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Every Halin graph is a Hamiltonian graph, and every edge of the graph belongs to a Hamiltonian cycle. Moreover, any Halin graph remains Hamiltonian after deletion of any vertex. Because every tree without vertices of degree 2 contains two leaves that share the same parent, every Halin graph contains a triangle. In particular, it is not possible for a Halin graph to be a triangle-free graph nor a bipartite graph.
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A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph. In graph theory, an odd cycle transversal of an undirected graph is a set of vertices of the graph that has a nonempty intersection with every odd cycle in the graph. Removing the vertices of an odd cycle transversal from a graph leaves a bipartite graph as the remaining induced subgraph.
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In the mathematical field of graph theory, the butterfly graph (also called the bowtie graph and the hourglass graph) is a planar undirected graph with 5 vertices and 6 edges.ISGCI: Information System on Graph Classes and their Inclusions. "List of Small Graphs". It can be constructed by joining 2 copies of the cycle graph C3 with a common vertex and is therefore isomorphic to the friendship graph F2.
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A -map graph is a map graph derived from a set of regions in which at most regions meet at any point. Equivalently, it is the half-square of a planar bipartite graph in which the vertex set (the side of the bipartition not used to induce the half-square) has maximum degree . A 3-map graph is a planar graph, and every planar graph can be represented as a 3-map graph. Every 4-map graph is a 1-planar graph, a graph that can be drawn with at most one crossing per edge, and every optimal 1-planar graph (a graph formed from a planar quadrangulation by adding two crossing diagonals to every quadrilateral face) is a 4-map graph.
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A Euclidean graph (a graph embedded in some Euclidean space) is periodic if there exists a basis of that Euclidean space whose corresponding translations induce symmetries of that graph (i.e., application of any such translation to the graph embedded in the Euclidean space leaves the graph unchanged). Equivalently, a periodic Euclidean graph is a periodic realization of an abelian covering graph over a finite graph. A Euclidean graph is uniformly discrete if there is a minimal distance between any two vertices.
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With DataScene, the user can plot 39 types 2D & 3D graphs (e.g., Area graph, Bar graph, Boxplot graph, Pie graph, Line graph, Histogram graph, Surface graph, Polar graph, Water Fall graph, etc.), manipulate, print, and export graphs to various formats (e.g., Bitmap, WMF/EMF, JPEG, PNG, GIF, TIFF, PostScript, and PDF), analyze data with different mathematical methods (fitting curves, calculating statics, FFT, etc.), create chart animations for presentations (e.g. with Powerpoint), classes, and web pages, and monitor and chart real- time data.
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In graph theory the circumference of a graph refers to the longest (simple) cycle contained in that graph.
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In graph theory, a dipole graph (also called a dipole or bond graph) is a multigraph consisting of two vertices connected with a number of parallel edges. A dipole graph containing n edges is called the order-n dipole graph, and is denoted by Dn. The order-n dipole graph is dual to the cycle graph Cn. The honeycomb as an abstract graph is the maximal abelian covering graph of the dipole graph D3, while the diamond crystal as an abstract graph is the maximal abelian covering graph of D4. Similarly to the Platonic graphs, the dipole graphs form the skeletons of the hosohedra. Their duals, the cycle graphs, form the skeletons of the dihedra.
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In the mathematical field of graph theory, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the Archimedean solids. It has 30 vertices and 60 edges, and is a quartic graph Archimedean graph.
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In the graph example, Graph cannot declare itself a friend Vertex. Rather, Vertex declares Graph a friend, and so provides Graph an access to its private fields. The fact that a class chooses its own friends means that friendship is not symmetric in general. In the graph example, Vertex cannot access private fields of Graph, although Graph can access private fields of Vertex.
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An example of a threshold graph. In graph theory, a threshold graph is a graph that can be constructed from a one-vertex graph by repeated applications of the following two operations: #Addition of a single isolated vertex to the graph. #Addition of a single dominating vertex to the graph, i.e. a single vertex that is connected to all other vertices.
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In the mathematical field of graph theory, the Brouwer-Haemers graph is a 20-regular undirected graph with 81 vertices and 810 edges. It is a strongly regular graph, a distance-transitive graph, and a Ramanujan graph. Although its construction is folklore, it was named after Andries Brouwer and Willem H. Haemers, who proved its uniqueness as a strongly regular graph.
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Every complete graph is a comparability graph, the comparability graph of a total order. All acyclic orientations of a complete graph are transitive. Every bipartite graph is also a comparability graph. Orienting the edges of a bipartite graph from one side of the bipartition to the other results in a transitive orientation, corresponding to a partial order of height two.
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The Paley graph of order 13, an example of a circulant graph. In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph, but this term has other meanings.
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Thus, the Shrikhande graph is a toroidal graph. The embedding forms a regular map in the torus, with 32 triangular faces. The skeleton of the dual of this map (as embedded in the torus) is the Dyck graph, a cubic symmetric graph. The Shrikhande graph is not a distance-transitive graph.
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In the mathematical field of graph theory, the Wagner graph is a 3-regular graph with 8 vertices and 12 edges. It is the 8-vertex Möbius ladder graph.
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The graph with only one vertex and no edges is called the trivial graph. A graph with only vertices and no edges is known as an edgeless graph. The graph with no vertices and no edges is sometimes called the null graph or empty graph, but the terminology is not consistent and not all mathematicians allow this object. Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable.
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In the mathematical field of graph theory, the Kittell graph is a planar graph with 23 vertices and 63 edges. Its unique planar embedding has 42 triangular faces. The Kittell graph is named after Irving Kittell, who used it as a counterexample to Alfred Kempe's flawed proof of the four-color theorem. Simpler counterexamples include the Errera graph and Poussin graph (both published earlier than Kittell) and the Fritsch graph and Soifer graph.
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As showed, this graph has boxicity exactly n; it is sometimes known as the Roberts graph. This graph is also the 1-skeleton of an n-dimensional cross-polytope; for instance, the graph T(6,3) = K2,2,2 is the octahedral graph, the graph of the regular octahedron. If n couples go to a party, and each person shakes hands with every person except his or her partner, then this graph describes the set of handshakes that take place; for this reason it is also called the cocktail party graph. The Turán graph T(n,2) is a complete bipartite graph and, when n is even, a Moore graph.
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The latter graph is the only line graph L(Kn,n) for which the strong regularity parameters do not determine that graph uniquely but are shared with a different graph, namely the Shrikhande graph (which is not a rook's graph).. The Shrikhande graph is locally hexagonal; that is, the neighbors of each vertex form a cycle of six vertices. As with any locally cyclic graph, the Shrikhande graph is the 1-skeleton of a Whitney triangulation of some surface; in the case of the Shrikhande graph, this surface is a torus in which each vertex is surrounded by six triangles.Brouwer, A. E. Shrikhande graph.
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In graph theory, a folded cube graph is an undirected graph formed from a hypercube graph by adding to it a perfect matching that connects opposite pairs of hypercube vertices.
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The medial graph of the Herschel graph is a 4-regular planar graph with no Hamiltonian decomposition. The shaded regions correspond to the vertices of the underlying Herschel graph. The Herschel graph also provides an example of a polyhedral graph for which the medial graph cannot be decomposed into two edge-disjoint Hamiltonian cycles. The medial graph of the Herschel graph is a 4-regular graph with 18 vertices, one for each edge of the Herschel graph; two vertices are adjacent in the medial graph whenever the corresponding edges of the Herschel graph are consecutive on one of its faces.. It is 4-vertex-connected and essentially 6-edge-connected, meaning that for every partition of the vertices into two subsets of at least two vertices, there are at least six edges crossing the partition.
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The smallest cubic graphs with crossing numbers 1–8 and 11 are known . The smallest 1-crossing cubic graph is the complete bipartite graph , with 6 vertices. The smallest 2-crossing cubic graph is the Petersen graph, with 10 vertices. The smallest 3-crossing cubic graph is the Heawood graph, with 14 vertices.
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Another conjecture of Berge, proved in 1972 by László Lovász, is that the complement of every perfect graph is also perfect. This became known as the perfect graph theorem, or (to distinguish it from the strong perfect graph conjecture/theorem) the weak perfect graph theorem. Because Berge's forbidden graph characterization is self-complementary, the weak perfect graph theorem follows immediately from the strong perfect graph theorem.
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A drawing of the Heawood graph with three crossings. This is the minimum number of crossings among all drawings of this graph, so the graph has crossing number . In graph theory, the crossing number of a graph is the lowest number of edge crossings of a plane drawing of the graph . For instance, a graph is planar if and only if its crossing number is zero.
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In graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges and vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K5 nor the complete bipartite graph K3,3., p. 77; .
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Unfortunately, it is difficult to modify a graph that is already running. It is usually easier to stop the graph and create a new graph from scratch. Starting with DirectShow 8.0, dynamic graph building, dynamic reconnection, and filter chains were introduced to help alter the graph while it was running. However, many filter vendors ignore this feature, making graph modification problematic after a graph has begun processing.
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A regular graph is a graph in which each vertex has the same number of neighbours, i.e., every vertex has the same degree. A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k.
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The alternative names "triangular graph". or "triangulated graph". have also been used, but are ambiguous, as they more commonly refer to the line graph of a complete graph and to the chordal graphs respectively. Every maximal planar graph is a least 3-connected.
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In the mathematical field of graph theory, the Livingstone graph is a distance-transitive graph with 266 vertices and 1463 edges. It is the largest distance-transitive graph with degree 11.
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The Frucht graph, one of the five smallest cubic graphs with no nontrivial graph automorphisms, is also a Halin graph.
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Besides being a specialized type of social graph, the enterprise social graph is related to network science and graph theory.
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An abstract graph is said to be n-apex if it can be made planar by deleting n or fewer vertices. A 1-apex graph is also said to be apex. According to , a graph is edge-apex if there is some edge in the graph that can be deleted to make the graph planar. A graph is contraction-apex if there is some edge in the graph that can be contracted to make the graph planar.
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Converting a triangle-free graph into a median graph. The problems of testing whether a graph is a median graph, and whether a graph is triangle-free, both had been well studied when observed that, in some sense, they are computationally equivalent.For previous median graph recognition algorithms, see , , and . For triangle detection algorithms, see , , and .
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The smallest 4-crossing cubic graph is the Möbius-Kantor graph, with 16 vertices. The smallest 5-crossing cubic graph is the Pappus graph, with 18 vertices. The smallest 6-crossing cubic graph is the Desargues graph, with 20 vertices. None of the four 7-crossing cubic graphs, with 22 vertices, are well known.
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A 3-coloring of a graph G may be described by a graph homomorphism from G to a triangle K3. In the language of homomorphisms, Grötzsch's theorem states that every triangle-free planar graph has a homomorphism to K3. Naserasr showed that every triangle-free planar graph also has a homomorphism to the Clebsch graph, a 4-chromatic graph. By combining these two results, it may be shown that every triangle-free planar graph has a homomorphism to a triangle-free 3-colorable graph, the tensor product of K3 with the Clebsch graph.
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In mathematics, the multi-level technique is a technique used to solve the graph partitioning problem. The idea of the multi-level technique is to reduce the magnitude of a graph by merging vertices together, compute a partition on this reduced graph, and finally project this partition on the original graph. In the first phase the magnitude of the graph is reduced by merging vertices. The merging of vertices is done iteratively: of a graph a new coarser graph is created and of this new coarser graph an even more coarse graph is created.
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In computer science, single pushout graph rewriting or SPO graph rewriting refers to a mathematical framework for graph rewriting, and is used in contrast to the double-pushout approach of graph rewriting.
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If every edge of a given graph G is subdivided, the resulting graph is a string graph if and only if G is planar. In particular, the subdivision of the complete graph K5 shown in the illustration is not a string graph, because K5 is not planar. Every circle graph, as an intersection graph of line segments (the chords of a circle), is also a string graph. Every chordal graph may be represented as a string graph: chordal graphs are intersection graphs of subtrees of trees, and one may form a string representation of a chordal graph by forming a planar embedding of the corresponding tree and replacing each subtree by a string that traces around the subtree's edges.
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A directed graph without directed cycles is called a directed acyclic graph. A connected graph without cycles is called a tree.
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The 110-vertex Iofinova-Ivanov graph is, in graph theory, a semi-symmetric cubic graph with 110 vertices and 165 edges.
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The polyhedral graph formed as the Schlegel diagram of a regular dodecahedron. Schlegel diagram of truncated icosidodecahedral graph In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected planar graphs.
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Colin de Verdière invariant is defined from a special class of matrices corresponding to a graph instead of just a single matrix related to the graph. Along the same lines other graph parameters are defined and studied, such as minimum rank of a graph, minimum semidefinite rank of a graph and minimum skew rank of a graph.
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The family of graphs in which each connected component is a cactus graph is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor. This minor is the diamond graph.. If both the butterfly graph and the diamond graph are forbidden minors, the family of graphs obtained is the family of pseudoforests.
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Mohar's research concerns topological graph theory, algebraic graph theory, graph minors, and graph coloring. With Carsten Thomassen he is the co-author of the book Graphs on Surfaces (Johns Hopkins University Press, 2001).
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Many natural and important concepts in graph theory correspond to other equally natural but different concepts in the dual graph. Because the dual of the dual of a connected plane graph is isomorphic to the primal graph,. each of these pairings is bidirectional: if concept in a planar graph corresponds to concept in the dual graph, then concept in a planar graph corresponds to concept in the dual.
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In the mathematical field of graph theory, the Wong graph is a 5-regular undirected graph with 30 vertices and 75 edges.. It is one of the four (5,5)-cage graphs, the others being the Foster cage, the Meringer graph, and the Robertson–Wegner graph. Like the unrelated Harries–Wong graph, it is named after Pak-Ken Wong.Wong, P. K. "Cages--A Survey." J. Graph Th. 6, 1-22, 1982.
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The web graph W4,2 is a cube. The web graph Wn,r is a graph consisting of r concentric copies of the cycle graph Cn, with corresponding vertices connected by "spokes". Thus Wn,1 is the same graph as Cn, and Wn,2 is a prism. A web graph has also been defined as a prism graph Yn+1, 3, with the edges of the outer cycle removed.
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The Heawood graph is the Levi graph of the Fano plane, the graph representing incidences between points and lines in that geometry. With this interpretation, the 6-cycles in the Heawood graph correspond to triangles in the Fano plane. Also, the Heawood graph is the Tits building of the group SL3(F2). The Heawood graph has crossing number 3, and is the smallest cubic graph with that crossing number .
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The smallest 8-crossing cubic graphs include the Nauru graph and the McGee graph or (3,7)-cage graph, with 24 vertices. The smallest 11-crossing cubic graphs include the Coxeter graph with 28 vertices. In 2009, Pegg and Exoo conjectured that the smallest cubic graph with crossing number 13 is the Tutte–Coxeter graph and the smallest cubic graph with crossing number 170 is the Tutte 12-cage.
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The Dürer graph is the graph formed by the vertices and edges of the Dürer solid. It is a cubic graph of girth 3 and diameter 4. As well as its construction as the skeleton of Dürer's solid, it can be obtained by applying a Y-Δ transform to the opposite vertices of a cube graph, or as the generalized Petersen graph G(6,2). As with any graph of a convex polyhedron, the Dürer graph is a 3-vertex-connected simple planar graph.
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While graph drawing and graph representation are valid topics in graph theory, in order to focus only on the abstract structure of graphs, a graph property is defined to be a property preserved under all possible isomorphisms of a graph. In other words, it is a property of the graph itself, not of a specific drawing or representation of the graph. Informally, the term "graph invariant" is used for properties expressed quantitatively, while "property" usually refers to descriptive characterizations of graphs. For example, the statement "graph does not have vertices of degree 1" is a "property" while "the number of vertices of degree 1 in a graph" is an "invariant".
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In the mathematical field of graph theory, a prism graph is a graph that has one of the prisms as its skeleton.
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In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G). The name line graph comes from a paper by although both and used the construction before this. Other terms used for the line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the ϑ-obrazom, as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph., p. 71.
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The Clebsch graph contains many copies of the Petersen graph as induced subgraphs: for each vertex v of the Clebsch graph, the ten non- neighbors of v induce a copy of the Petersen graph.
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In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges connecting pairs of vertices in that subset.
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In the mathematical field of graph theory, the Hoffman graph is a 4-regular graph with 16 vertices and 32 edges discovered by Alan Hoffman. Published in 1963, it is cospectral to the hypercube graph Q4.Hoffman, A. J. "On the Polynomial of a Graph." Amer. Math.
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A signed graph is a special kind of gain graph, where the gain group has order 2. The pair (G, B(Σ)) determined by a signed graph Σ is a special kind of biased graph.
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In graph theory, a highly irregular graph is a graph in which, for every vertex, all neighbors of that vertex have distinct degrees.
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Unary operations create a new graph from a single initial graph.
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If G is a 3-regular triangle-free graph, then the line graph L(G) is a graph formed by expanding each edge of G into a new vertex, and making two vertices adjacent in L(G) when the corresponding edges of G share an endpoint. These graphs are 4-regular and locally linear. Every 4-regular locally linear graph can be constructed in this way. For instance, the graph of the cuboctahedron can be formed in this way as the line graph of a cube, and the nine-vertex Paley graph is the line graph of the utility graph K_{3,3}.
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In the mathematical field of graph theory, the Grötzsch graph is a triangle- free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5. It is named after German mathematician Herbert Grötzsch. The Grötzsch graph is a member of an infinite sequence of triangle-free graphs, each the Mycielskian of the previous graph in the sequence, starting from the null graph; this sequence of graphs was used by to show that there exist triangle- free graphs with arbitrarily large chromatic number. Therefore, the Grötzsch graph is sometimes also called the Mycielski graph or the Mycielski–Grötzsch graph.
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A triangular grid graph is a graph that corresponds to a triangular grid. A Hanan grid graph for a finite set of points in the plane is produced by the grid obtained by intersections of all vertical and horizontal lines through each point of the set. The rook's graph (the graph that represents all legal moves of the rook chess piece on a chessboard) is also sometimes called the lattice graph, although this graph is strictly different than the lattice graph described in this article. The valid moves of fairy chess piece wazir form the square lattice graph.
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The characteristic polynomial of the 5-regular Clebsch graph is (x+3)^5(x-1)^{10}(x-5). Because this polynomial can be completely factored into linear terms with integer coefficients, the Clebsch graph is an integral graph: its spectrum consists entirely of integers. The Clebsch graph is the only graph with this characteristic polynomial, making it a graph determined by its spectrum. The 5-regular Clebsch graph is a Cayley graph with an automorphism group of order 1920, isomorphic to the Coxeter group D_5.
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In mathematics, a universal graph is an infinite graph that contains every finite (or at-most-countable) graph as an induced subgraph. A universal graph of this type was first constructed by Richard Rado and is now called the Rado graph or random graph. More recent work has focused on universal graphs for a graph family : that is, an infinite graph belonging to F that contains all finite graphs in . For instance, the Henson graphs are universal in this sense for the -clique-free graphs.
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In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring. Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by .
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The graph of vertices and edges of the 3-3 duoprism has 9 vertices and 18 edges. Like the Berlekamp–van Lint–Seidel graph and the unknown solution to Conway's 99-graph problem, every edge is part of a unique triangle and every non-adjacent pair of vertices is the diagonal of a unique square. It is a toroidal graph, a locally linear graph, a strongly regular graph with parameters (9,4,1,2), the 3\times 3 rook's graph, and the Paley graph of order 9.
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A Ptolemaic graph The gem graph or 3-fan is not Ptolemaic. A block graph, a special case of a Ptolemaic graph Three operations by which any distance- hereditary graph can be constructed. For Ptolemaic graphs, the neighbors of false twins are restricted to form a clique, preventing the construction of the 4-cycle shown here. In graph theory, a Ptolemaic graph is an undirected graph whose shortest path distances obey Ptolemy's inequality, which in turn was named after the Greek astronomer and mathematician Ptolemy.
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Any graph which can be embedded in a plane can also be embedded in a torus. A toroidal graph of genus 1 can be embedded in a torus but not in a plane. The Heawood graph, the complete graph K7 (and hence K5 and K6), the Petersen graph (and hence the complete bipartite graph K3,3, since the Petersen graph contains a subdivision of it), one of the Blanuša snarks, and all Möbius ladders are toroidal. More generally, any graph with crossing number 1 is toroidal.
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A graph is a parity graph if and only if every component of its split decomposition is either a complete graph or a bipartite graph. Based on this characterization, it is possible to test whether a given graph is a parity graph in linear time. The same characterization also leads to generalizations of some graph optimization algorithms from bipartite graphs to parity graphs. For instance, using the split decomposition, it is possible to find the weighted maximum independent set of a parity graph in polynomial time..
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A citation graph having vertices representing the papers in the 1994–2000 Graph Drawing symposia and having edges representing citations between these papers was made available as part of the graph drawing contest associated with the 2001 symposium.. The largest connected component of this graph consists of 249 vertices and 642 edges; clustering analysis reveals several prominent subtopics within graph drawing that are more tightly connected, including three-dimensional graph drawing and orthogonal graph drawing..
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The graph is cubic, and all cycles in the graph have six or more edges. Every smaller cubic graph has shorter cycles, so this graph is the 6-cage, the smallest cubic graph of girth 6. It is a distance- transitive graph (see the Foster census) and therefore distance regular. There are 24 perfect matchings in the Heawood graph; for each matching, the set of edges not in the matching forms a Hamiltonian cycle.
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Both Tietze's graph and the Petersen graph are maximally nonhamiltonian: they have no Hamiltonian cycle, but any two non- adjacent vertices can be connected by a Hamiltonian path. Tietze's graph and the Petersen graph are the only 2-vertex-connected cubic non-Hamiltonian graphs with 12 or fewer vertices. Unlike the Petersen graph, Tietze's graph is not hypohamiltonian: removing one of its three triangle vertices forms a smaller graph that remains non-Hamiltonian.
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The graph C is a planar cover of the graph H. The covering map is indicated by the vertex colors. In graph theory, a planar cover of a finite graph G is a finite covering graph of G that is itself a planar graph. Every graph that can be embedded into the projective plane has a planar cover; an unsolved conjecture of Seiya Negami states that these are the only graphs with planar covers., p.
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A maximal outerplanar graph and its 3-coloring. The complete graph K4 is the smallest planar graph that is not outerplanar. In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two forbidden minors and , or by their Colin de Verdière graph invariants.
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In the mathematical field of graph theory, the Frucht graph is a 3-regular graph with 12 vertices, 18 edges, and no nontrivial symmetries. It was first described by Robert Frucht in 1939. The Frucht graph is a pancyclic Halin graph with chromatic number 3, chromatic index 3, radius 3, and diameter 4. As with every Halin graph, the Frucht graph is polyhedral (planar and 3-vertex- connected) and Hamiltonian, with girth 3.
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A connected component of the undirected subgraph of a chain graph is called a chain. A chain graph may be transformed into an undirected graph by constructing its moral graph, an undirected graph formed from the chain graph by adding undirected edges between pairs of vertices that have outgoing edges to the same chain, and then forgetting the orientations of the directed edges.
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In computational biology, power graph analysis is a method for the analysis and representation of complex networks. Power graph analysis is the computation, analysis and visual representation of a power graph from a graph (networks). Power graph analysis can be thought of as a lossless compression algorithm for graphs. It extends graph syntax with representations of cliques, bicliques and stars.
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In graph theory, a graph amalgamation is a relationship between two graphs (one graph is an amalgamation of another). Similar relationships include subgraphs and minors. Amalgamations can provide a way to reduce a graph to a simpler graph while keeping certain structure intact. The amalgamation can then be used to study properties of the original graph in an easier to understand context.
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An aperiodic graph. The cycles in this graph have lengths 5 and 6; therefore, there is no k > 1 that divides all cycle lengths. strongly connected graph with period three. In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer k > 1 that divides the length of every cycle of the graph.
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They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices. The complement graph of a complete graph is an empty graph. If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. Kn can be decomposed into trees Ti such that Ti has vertices.
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It is the smallest distance-regular graph that is not distance-transitive.. The automorphism group of the Shrikhande graph is of order 192. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Shrikhande graph is a symmetric graph. The characteristic polynomial of the Shrikhande graph is : (x-6)(x-2)^6(x+2)^9.
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An example of how intersecting sets defines a graph. In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types of sets that are used to form an intersection representation of them.
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The column-store database offers graph database capabilities. The graph engine processes the Cypher Query Language and also has a visual graph manipulation via a tool called Graph Viewer. Graph data structures are stored directly in relational tables in HANA's column store. Pre-built algorithms in the graph engine include pattern matching, neighborhood search, single shortest path, and strongly connected components.
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In the mathematical field of graph theory, the diamond graph is a planar undirected graph with 4 vertices and 5 edges.ISGCI: Information System on Graph Classes and their Inclusions "List of Small Graphs". It consists of a complete graph K_4 minus one edge. The diamond graph has radius 1, diameter 2, girth 3, chromatic number 3 and chromatic index 3\.
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In graph theory, the girth of a graph is the length of a shortest cycle contained in the graph.R. Diestel, Graph Theory, p.8. 3rd Edition, Springer- Verlag, 2005 If the graph does not contain any cycles (i.e. it's an acyclic graph), its girth is defined to be infinity.
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A self-complementary graph: the blue N is isomorphic to its complement, the dashed red Z. A self-complementary graph is a graph which is isomorphic to its complement. The simplest non-trivial self-complementary graphs are the 4-vertex path graph and the 5-vertex cycle graph.
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Other equivalent representations for cellular embeddings include the ribbon graph, a topological space formed by gluing together topological disks for the vertices and edges of an embedded graph, and the graph-encoded map, an edge-colored cubic graph with four vertices for each edge of the embedded graph.
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1-factorization of Desargues graph: each color class is a 1-factor. Petersen graph can be partitioned into a 1-factor (red) and a 2-factor (blue). However, the graph is not 1-factorable. In graph theory, a factor of a graph G is a spanning subgraph, i.e.
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In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges, the smallest non-Hamiltonian polyhedral graph. It is named after British astronomer Alexander Stewart Herschel.
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The former is the colinearity graph of the generalized quadrangle GQ(3,9). The latter is a strongly regular graph called the local McLaughlin graph.
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In graph theory, a voltage graph is a directed graph whose edges are labelled invertibly by elements of a group. It is formally identical to a gain graph, but it is generally used in topological graph theory as a concise way to specify another graph called the derived graph of the voltage graph. Typical choices of the groups used for voltage graphs include the two-element group ℤ2 (for defining the bipartite double cover of a graph), free groups (for defining the universal cover of a graph), d-dimensional integer lattices ℤd (viewed as a group under vector addition, for defining periodic structures in d-dimensional Euclidean space),; ; . and finite cyclic groups ℤn for n > 2\.
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This definition produces the same answer, n − 1, for the connectivity of the complete graph Kn. A 1-connected graph is called connected; a 2-connected graph is called biconnected. A 3-connected graph is called triconnected.
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Sometimes the term "ladder graph" is used for the n × P2 ladder rung graph, which is the graph union of n copies of the path graph P2. The ladder rung graphs LR1, LR2, LR3, LR4, and LR5.
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The two others are cycle graph C5 and the complete graph K5.
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If the chromatic number of a graph is uncountable, then the graph necessarily contains as a subgraph a half graph on the natural numbers. This half graph, in turn, contains every complete bipartite graph in which one side of the bipartition is finite and the other side is countably infinite.
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58 Graph edit distance is one of similarity measures suggested for graph matching.Bridging the Gap Between Graph Edit Distance and Kernel Machines, p. 16Horst Bunke, Xiaoyi Jang, "Graph Matching and Similarity", in: Intelligent Systems and Interfaces, pp. 281-304 (2000) The class of algorithms is called error-tolerant graph matching.
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When plotted as a graph, the lettered vertices are sequentially connected by edges to spell a word. If the graph is non-planar, the word is an eodermdrome. The graph of eodermdrome is the non-planar graph K5. K5 graph of eodermdrome Eckler searched for all eodermdromes in Webster's Dictionary.
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Construction of two demicubes (regular tetrahedra, forming a stella octangula) from a single cube. The halved cube graph of order three is the graph of vertices and edges of a single demicube. The halved cube graph of order four includes all of the cube vertices and edges, and all of the edges of the two demicubes. In graph theory, the halved cube graph or half cube graph of order n is the graph of the demihypercube, formed by connecting pairs of vertices at distance exactly two from each other in the hypercube graph.
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The Petersen graph (on the left) and its complement graph (on the right). In graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there.. It is not, however, the set complement of the graph; only the edges are complemented.
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The red graph is the dual graph of the blue graph, and vice versa. In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge whenever two faces of are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge of has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of .
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Clique percolation methods may be generalized by recording different amounts of overlap between the various k-cliques. This then defines a new type of graph, a clique graph, where each k-clique in the original graph is represented by a vertex in the new clique graph. The edges in the clique graph are used to record the strength of the overlap of cliques in the original graph. One may then apply any community detection method to this clique graph to identify the clusters in the original graph through the k-clique structure.
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Every cycle graph is a circulant graph, as is every crown graph with vertices. The Paley graphs of order (where is a prime number congruent to ) is a graph in which the vertices are the numbers from 0 to and two vertices are adjacent if their difference is a quadratic residue modulo . Since the presence or absence of an edge depends only on the difference modulo of two vertex numbers, any Paley graph is a circulant graph. Every Möbius ladder is a circulant graph, as is every complete graph.
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In the mathematical field of graph theory, the Holt graph or Doyle graph is the smallest half-transitive graph, that is, the smallest example of a vertex- transitive and edge-transitive graph which is not also symmetric.Doyle, P. "A 27-Vertex Graph That Is Vertex-Transitive and Edge-Transitive But Not L-Transitive." October 1998. . Such graphs are not common.Jonathan L. Gross, Jay Yellen, Handbook of Graph Theory, CRC Press, 2004, , p. 491. It is named after Peter G. Doyle and Derek F. Holt, who discovered the same graph independently in 1976.
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A subdivision of K3,3 in the generalized Petersen graph G(9,2), showing that the graph is nonplanar. In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (the complete graph on five vertices) or of K3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three, also known as the utility graph).
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In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood.
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A thagomizer graph In a 2017 paper, the term "thagomizer graph" (and also the associated "thagomizer matroid") has been introduced for the complete tripartite graph .
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The graph sandwich problem is NP-complete when Π is the property of being a chordal graph, comparability graph, permutation graph, chordal bipartite graph, or chain graph.. It can be solved in polynomial time for split graphs,. threshold graphs, and graphs in which every five vertices contain at most one four- vertex induced path.. The complexity status has also been settled for the H-free graph sandwich problems for each of the four-vertex graphs H..
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Among Fan Chung's publications, her contributions to spectral graph theory are important to this area of graph theory. From the first publications about undirected graphs to recent publications about the directed graphs, Fan Chung creates the solid base in the spectral graph theory to the future graph theorist. Spectral graph theory, as one of the most important theories in graph theory, combines the algebra and graph perfectly. Historically, algebraic methods treat many types of graphs efficiently.
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Such rules consist of an original graph, which is to be matched to a subgraph in the complete state, and a replacing graph, which will replace the matched subgraph. Formally, a graph rewriting system usually consists of a set of graph rewrite rules of the form L \rightarrow R, with L being called pattern graph (or left-hand side) and R being called replacement graph (or right-hand side of the rule). A graph rewrite rule is applied to the host graph by searching for an occurrence of the pattern graph (pattern matching, thus solving the subgraph isomorphism problem) and by replacing the found occurrence by an instance of the replacement graph. Rewrite rules can be further regulated in the case of labeled graphs, such as in string-regulated graph grammars.
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Every comparability graph is perfect: this is essentially just Mirsky's theorem, restated in graph-theoretic terms . By the perfect graph theorem of , the complement of any perfect graph is also perfect. Therefore, the complement of any comparability graph is perfect; this is essentially just Dilworth's theorem itself, restated in graph-theoretic terms . Thus, the complementation property of perfect graphs can provide an alternative proof of Dilworth's theorem.
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Every distance-hereditary graph is also a parity graph, a graph in which every two induced paths between the same pair of vertices both have odd length or both have even length., p.45. Every even power of a distance-hereditary graph G (that is, the graph G2i formed by connecting pairs of vertices at distance at most 2i in G) is a chordal graph., Theorem 10.6.
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Graph databases are a powerful tool for graph-like queries. For example, computing the shortest path between two nodes in the graph. Other graph-like queries can be performed over a graph database in a natural way (for example graph's diameter computations or community detection). Graphs are flexible, meaning it allows the user to insert new data into the existing graph without loss of application functionality.
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A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles. In graph theory, a line perfect graph is a graph whose line graph is a perfect graph. Equivalently, these are the graphs in which every odd-length simple cycle is a triangle.
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A plane graph (in blue) and its directed medial graph (in red). The medial graph definition can be extended to include an orientation. First, the faces of the medial graph are colored black if they contain a vertex of the original graph and white otherwise. This coloring causes each edge of the medial graph to be bordered by one black face and one white face.
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The Grötzsch graph is a triangle-free graph that cannot be colored with fewer than four colors Much research about triangle-free graphs has focused on graph coloring. Every bipartite graph (that is, every 2-colorable graph) is triangle-free, and Grötzsch's theorem states that every triangle-free planar graph may be 3-colored.; ). However, nonplanar triangle-free graphs may require many more than three colors.
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The number of components is an important topological invariant of a graph. In topological graph theory it can be interpreted as the zeroth Betti number of the graph. In algebraic graph theory it equals the multiplicity of 0 as an eigenvalue of the Laplacian matrix of the graph. It is also the index of the first nonzero coefficient of the chromatic polynomial of a graph.
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The butterfly graph (left) and diamond graph (right), forbidden minors for pseudoforests Forming a minor of a pseudoforest by contracting some of its edges and deleting others produces another pseudoforest. Therefore, the family of pseudoforests is closed under minors, and the Robertson–Seymour theorem implies that pseudoforests can be characterized in terms of a finite set of forbidden minors, analogously to Wagner's theorem characterizing the planar graphs as the graphs having neither the complete graph K5 nor the complete bipartite graph K3,3 as minors. As discussed above, any non-pseudoforest graph contains as a subgraph a handcuff, figure 8, or theta graph; any handcuff or figure 8 graph may be contracted to form a butterfly graph (five-vertex figure 8), and any theta graph may be contracted to form a diamond graph (four-vertex theta graph),For this terminology, see the list of small graphs from the Information System on Graph Class Inclusions. However, butterfly graph may also refer to a different family of graphs related to hypercubes, and the five-vertex figure 8 is sometimes instead called a bowtie graph.
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The automorphism group of the Petersen graph is the symmetric group S_5; the action of S_5 on the Petersen graph follows from its construction as a Kneser graph. Every homomorphism of the Petersen graph to itself that doesn't identify adjacent vertices is an automorphism. As shown in the figures, the drawings of the Petersen graph may exhibit five-way or three-way symmetry, but it is not possible to draw the Petersen graph in the plane in such a way that the drawing exhibits the full symmetry group of the graph. Despite its high degree of symmetry, the Petersen graph is not a Cayley graph.
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A directed graph with three vertices (blue circles) and three edges (black arrows). In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points), together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. These pairs are known as edges (also called links or lines), and for a directed graph are also known as arrows.
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The neighborhood of any vertex in the Schläfli graph forms a 16-vertex subgraph in which each vertex has 10 neighbors (the numbers 16 and 10 coming from the parameters of the Schläfli graph as a strongly regular graph). These subgraphs are all isomorphic to the complement graph of the Clebsch graph.. Note that Cameron and van Lint use an alternative definition of these graphs in which both the Schläfli graph and the Clebsch graph are complemented from their definitions here. Since the Clebsch graph is triangle-free, the Schläfli graph is claw-free. It plays an important role in the structure theory for claw-free graphs by .
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K5 (left) and K3,3 (right) as minors of the nonplanar Petersen graph (small colored circles and solid black edges). The minors may be formed by deleting the red vertex and contracting edges within each yellow circle. A clique-sum of two planar graphs and the Wagner graph, forming a K5-free graph. In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither K5 (the complete graph on five vertices) nor K3,3 (the utility graph, a complete bipartite graph on six vertices).
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In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is an embedding with the property that every cycle is the boundary of a topological disk whose interior is disjoint from the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three- dimensional analogue of the planar graphs.. Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding. Flat embeddings are automatically linkless, but not vice versa.
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The bipartite Kneser graph can be formed as a bipartite double cover of in which one makes two copies of each vertex and replaces each edge by a pair of edges connecting corresponding pairs of vertices . The bipartite Kneser graph is the Desargues graph and the bipartite Kneser graph is a crown graph.
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Given a graph, deciding whether it is the square of another graph is NP-complete. . Moreover, it is NP-complete to determine whether a graph is a kth power of another graph, for a given number k ≥ 2, or whether it is a kth power of a bipartite graph, for k > 2..
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In an undirected graph, an unordered pair of vertices is called connected if a path leads from x to y. Otherwise, the unordered pair is called disconnected. A connected graph is an undirected graph in which every unordered pair of vertices in the graph is connected. Otherwise, it is called a disconnected graph.
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In the mathematical field of graph theory, the Schläfli graph, named after Ludwig Schläfli, is a 16-regular undirected graph with 27 vertices and 216 edges. It is a strongly regular graph with parameters srg(27, 16, 10, 8).
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Similar to classical noise reduction of signals based on Fourier transform, graph filters based on the Graph Fourier transform can be designed for graph signal denoising.
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This graph family may be characterized by a single forbidden minor, the four-vertex diamond graph formed by removing an edge from the complete graph K4.
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In graph theory, the Poussin graph is a planar graph with 15 vertices and 39 edges. It is named after Charles Jean de la Vallée-Poussin.
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An asymmetric graph is a graph for which there are no other automorphisms.
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The term barycentric division is also used in graph theory (Barycentric_Subdivision (Graph Theory)).
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For editing a graph, Gnumeric displays a window where all the elements of the graph are listed. Other spreadsheet programs typically require the user to select the individual elements of the graph in the graph itself in order to edit them.
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Wolz, Jessica; Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018 The Dyck graph is a toroidal graph, and the dual of its symmetric toroidal embedding is the Shrikhande graph, a strongly regular graph both symmetric and hamiltonian.
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The first counterexample to the Tutte conjecture was the Horton graph, published by . After the Horton graph, a number of smaller counterexamples to the Tutte conjecture were found. Among them are a 92-vertex graph by ,. a 78-vertex graph by ,.
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On can also say that G is permutationally k-representable. A graph is permutationally representable iff it is a comparability graph . A graph is word-representable implies that the neighbourhood of each vertex is permutationally representable (i.e. is a comparability graph) .
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In computer science, graph transformation, or graph rewriting, concerns the technique of creating a new graph out of an original graph algorithmically. It has numerous applications, ranging from software engineering (software construction and also software verification) to layout algorithms and picture generation. Graph transformations can be used as a computation abstraction. The basic idea is that if the state of a computation can be represented as a graph, further steps in that computation can then be represented as transformation rules on that graph.
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Adding a name to the triple makes a "quad store" or named graph. A graph database has a more generalized structure than a triplestore, using graph structures with nodes, edges, and properties to represent and store data. Graph databases might provide index-free adjacency, meaning every element contains a direct pointer to its adjacent elements, and no index lookups are necessary. General graph databases that can store any graph are distinct from specialized graph databases such as triplestores and network databases.
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Wheel graphs are planar graphs, and as such have a unique planar embedding. More specifically, every wheel graph is a Halin graph. They are self-dual: the planar dual of any wheel graph is an isomorphic graph. Every maximal planar graph, other than K4 = W4, contains as a subgraph either W5 or W6. There is always a Hamiltonian cycle in the wheel graph and there are n^2-3n+3 cycles in Wn . The 7 cycles of the wheel graph W4.
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A claw In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph K1,3 (that is, a star graph with three edges, three leaves, and one central vertex). A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern.
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In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v1 and v2 of G, there is some automorphism :f\colon G \to G\ such that :f(v_1) = v_2.\ In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices.. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph and Tietze's graph).
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A symmetric directed graph (that is, a directed graph in which the reverse of every edge is also an edge) is sometimes also called a "bidirected graph".
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It admits the graph as fundamental domain. The graph of groups given by the stabiliser subgroups on the fundamental domain corresponds to the original graph of groups.
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An outer-1-planar graph, analogously to 1-planar graphs can be drawn in a disk, with the vertices on the boundary of the disk, and with at most one crossing per edge. Every maximal outerplanar graph is a chordal graph. Every maximal outerplanar graph is the visibility graph of a simple polygon.; ; , Theorem 4.10.
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In mathematics, graph Fourier transform is a mathematical transform which eigendecomposes the Laplacian matrix of a graph into eigenvalues and eigenvectors. Analogously to classical Fourier Transform, the eigenvalues represent frequencies and eigenvectors form what is known as a graph Fourier basis. The Graph Fourier transform is important in spectral graph theory. It is widely applied in the recent study of graph structured learning algorithms, such as the widely employed convolutional networks.
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Node graph architecture is a type of software design which builds around modular node components which can be connected to form a graph. Often the software's underlying node graph architecture is also exposed to the end user as a two-dimensional visualization of the node graph. The node graph architecture is popular in the film and video game industry. There are often many different node types participating in the node graph.
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Cycles of all possible lengths in the graph of an octahedron, showing it to be pancyclic. In the mathematical study of graph theory, a pancyclic graph is a directed graph or undirected graph that contains cycles of all possible lengths from three up to the number of vertices in the graph.. Pancyclic graphs are a generalization of Hamiltonian graphs, graphs which have a cycle of the maximum possible length.
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A graph is trivially a quotient graph of itself (each block of the partition is a single vertex), and the graph consisting of a single point is the quotient graph of any non-empty graph (the partition consisting of a single block of all vertices). The simplest non-trivial quotient graph is one obtained by identifying two vertices (vertex identification); if the vertices are connected, this is called edge contraction.
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In the mathematical field of graph theory, the Foster cage is a 5-regular undirected graph with 30 vertices and 75 edges.. It is one of the four (5,5)-cage graphs, the others being the Meringer graph, the Robertson–Wegner graph, and the Wong graph. Like the unrelated Foster graph, it is named after R. M. Foster. It has chromatic number 4, diameter 3, and is 5-vertex- connected.
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The smallest possible number of vertices for a non-hamiltonian polyhedral graph is 11. Therefore, the Goldner–Harary graph is a minimal example of graphs of this type. However, the Herschel graph, another non-Hamiltonian polyhedron with 11 vertices, has fewer edges. As a non-Hamiltonian maximal planar graph, the Goldner–Harary graph provides an example of a planar graph with book thickness greater than two.. See in particular Figure 9.
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Two polytopes are called combinatorially isomorphic if their face lattices are isomorphic. The polytope graph (polytopal graph, graph of the polytope, 1-skeleton) is the set of vertices and edges of the polytope only, ignoring higher-dimensional faces. For instance, a polyhedral graph is the polytope graph of a three-dimensional polytope. By a result of Whitney the face lattice of a three-dimensional polytope is determined by its graph.
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Graph traversal is a subroutine in most graph algorithms. The goal of a graph traversal algorithm is to visit (and / or process) every node of a graph. Graph traversal algorithms, like breadth-first search and depth-first search, are analyzed using the von Neumann model, which assumes uniform memory access cost. This view neglects the fact, that for huge instances part of the graph resides on disk rather than internal memory.
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A 3-edge-coloring of the Desargues graph. In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring.
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In graph theory, the perfect graph theorem of states that an undirected graph is perfect if and only if its complement graph is also perfect. This result had been conjectured by , and it is sometimes called the weak perfect graph theorem to distinguish it from the strong perfect graph theoremThis was also conjectured by Berge but only proven much later by . characterizing perfect graphs by their forbidden induced subgraphs.
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This graph model enables an implementation of ZF without infinity as data types and thus an interpretation of set theory in expressive type theories. Graph models exist for ZF and also set theories different from Zermelo set theory, such as non-well founded theories. Such models have more intricate edge structure. In graph theory, the graph whose vertices correspond to hereditarily finite sets is the Rado graph or random graph.
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In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function between topological spaces has a closed graph if its graph is a closed subset of the product space . A related property is open graph. This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous.
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In computer science, double pushout graph rewriting (or DPO graph rewriting) refers to a mathematical framework for graph rewriting. It was introduced as one of the first algebraic approaches to graph rewriting in the article "Graph-grammars: An algebraic approach" (1973)."Graph-grammars: An algebraic approach", Ehrig, Hartmut and Pfender, Michael and Schneider, Hans-Jürgen, Switching and Automata Theory, 1973. SWAT'08. IEEE Conference Record of 14th Annual Symposium on, pp.
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An apex graph is a graph that may be made planar by the removal of one vertex, and a k-apex graph is a graph that may be made planar by the removal of at most k vertices. A 1-planar graph is a graph that may be drawn in the plane with at most one simple crossing per edge, and a k-planar graph is a graph that may be drawn with at most k simple crossings per edge. A map graph is a graph formed from a set of finitely many simply- connected interior-disjoint regions in the plane by connecting two regions when they share at least one boundary point. When at most three regions meet at a point, the result is a planar graph, but when four or more regions meet at a point, the result can be nonplanar.
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Parthasarathy wrote a book on graph theory, Basic Graph Theory (Tata McGraw-Hill, 1994).
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The Cameron graph is a strongly regular graph of parameters (231, 30, 9, 3).
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With this construction, one can embed the Gewirtz graph in the Higman–Sims graph.
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In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the groups will produce edges in the partitioned graph. If the number of resulting edges is small compared to the original graph, then the partitioned graph may be better suited for analysis and problem-solving than the original. Finding a partition that simplifies graph analysis is a hard problem, but one that has applications to scientific computing, VLSI circuit design, and task scheduling in multiprocessor computers, among others.
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See , Fact 1 and its proof. This result, shown by , justifies the definite article in the common alternative name "the random graph" for the Rado graph. Repeatedly drawing a finite graph from the Erdős–Rényi model will in general lead to different graphs; however, when applied to a countably infinite graph, the model almost always produces the same infinite graph. For any graph generated randomly in this way, the complement graph can be obtained at the same time by reversing all the choices: including an edge when the first graph did not include the same edge, and vice versa.
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Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces. However, these notions of dual graphs should not be confused with a different notion, the edge-to-vertex dual or line graph of a graph. The term dual is used because the property of being a dual graph is symmetric, meaning that if is a dual of a connected graph , then is a dual of . When discussing the dual of a graph , the graph itself may be referred to as the "primal graph".
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Graph matching is the problem of finding a similarity between graphs.Endika Bengoetxea, "Inexact Graph Matching Using Estimation of Distribution Algorithms", Ph. D., 2002, Chapter 2:The graph matching problem (retrieved June 28, 2017) Graphs are commonly used to encode structural information in many fields, including computer vision and pattern recognition, and graph matching is an important tool in these areas. Endika Bengoetxea, Ph.D., Abstract In these areas it is commonly assumed that the comparison is between the data graph and the model graph. The case of exact graph matching is known as the graph isomorphism problem.
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A graph is line perfect if and only if each of its biconnected components is a bipartite graph, the complete graph K_4, or a triangular book K_{1,1,n}. Because these three types of biconnected component are all perfect graphs themselves, every line perfect graph is itself perfect. By similar reasoning, every line perfect graph is a parity graph, a Meyniel graph, and a perfectly orderable graph. Line perfect graphs generalize the bipartite graphs, and share with them the properties that the maximum matching and minimum vertex cover have the same size, and that the chromatic index equals the maximum degree.
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It is NP-hard to test whether a given undirected planar graph can be realized as a matchstick graph... More precisely, this problem is complete for the existential theory of the reals.. provides some easily tested necessary criteria for a graph to be a matchstick graph, but these are not also sufficient criteria: a graph may pass Kurz's tests and still not be a matchstick graph.. It is also NP-complete to determine whether a matchstick graph has a Hamiltonian cycle, even when the vertices of the graph all have integer coordinates that are given as part of the input to the problem..
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In graph theory, a moral graph is used to find the equivalent undirected form of a directed acyclic graph. It is a key step of the junction tree algorithm, used in belief propagation on graphical models. The moralized counterpart of a directed acyclic graph is formed by adding edges between all pairs of non- adjacent nodes that have a common child, and then making all edges in the graph undirected. Equivalently, a moral graph of a directed acyclic graph is an undirected graph in which each node of the original is now connected to its Markov blanket.
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A helm graph, denoted Hn is a graph obtained by attaching a single edge and node to each node of the outer circuit of a wheel graph Wn.
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And it is both a complete graph and an edgeless graph. However, definitions for each of these graph properties will vary depending on whether context allows for K_0.
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A minimal imperfect graph is a graph in which every subgraph is perfect. The deletion of any vertex from a minimal imperfect graph leaves a uniquely colorable subgraph.
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A map graph (top), the cocktail party graph K2,2,2,2, defined by corner adjacency of eight regions in the plane (lower left), or as the half-square of a planar bipartite graph (lower right, the graph of the rhombic dodecahedron) In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally disjoint regions of the Euclidean plane. The map graphs include the planar graphs, but are more general. Any number of regions can meet at a common corner (as in the Four Corners of the United States, where four states meet), and when they do the map graph will contain a clique connecting the corresponding vertices, unlike planar graphs in which the largest cliques have only four vertices.. Another example of a map graph is the king's graph, a map graph of the squares of the chessboard connecting pairs of squares between which the chess king can move.
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The poster session at Graph Drawing 2009 in Chicago. The International Symposium on Graph Drawing (GD) is an annual academic conference in which researchers present peer reviewed papers on graph drawing, information visualization of network information, geometric graph theory, and related topics.
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The transformation of graphs is often formalized and represented by graph rewrite systems. Complementary to graph transformation systems focusing on rule-based in-memory manipulation of graphs are graph databases geared towards transaction-safe, persistent storing and querying of graph-structured data.
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Such a graph may be made into a strongly connected graph by adding one more edge, from the sink back to the source, through the outer face. The dual of this augmented planar graph is itself the augmentation of another st-planar graph.
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A hypohamiltonian graph constructed by . In the mathematical field of graph theory, a graph G is said to be hypohamiltonian if G does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from G is Hamiltonian.
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A complete graph with five vertices and ten edges. Each vertex has an edge to every other vertex. A complete graph is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges.
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In the study of graph algorithms, an implicit graph representation (or more simply implicit graph) is a graph whose vertices or edges are not represented as explicit objects in a computer's memory, but rather are determined algorithmically from some more concise input.
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The graph realization problem is a decision problem in graph theory. Given a finite sequence (d_1,\dots,d_n) of natural numbers, the problem asks whether there is labeled simple graph such that (d_1,\dots,d_n) is the degree sequence of this graph.
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As the Graph Fourier transform enables the definition of convolution on graphs, it makes possible to adapt the conventional convolutional neural networks (CNN) to work on graphs. Graph structured semi-supervised learning algorithms such as graph convolutional network (GCN), are able to propagate the labels of a graph signal throughout the graph with a small subset of labelled nodes.
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A 14-vertex half graph In graph theory, a branch of mathematics, a half graph is a special type of bipartite graph. These graphs are called the half graphs because they have approximately half of the edges of a complete bipartite graph on the same vertices. The name was given to these graphs by Paul Erdős and András Hajnal.
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In the mathematical field of graph theory, the Biggs-Smith graph is a 3-regular graph with 102 vertices and 153 edges. It has chromatic number 3, chromatic index 3, radius 7, diameter 7 and girth 9. It is also a 3-vertex- connected graph and a 3-edge-connected graph. All the cubic distance-regular graphs are known.
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An undirected graph At its simplest, DOT can be used to describe an undirected graph. An undirected graph shows simple relations between objects, such as friendship between people. The graph keyword is used to begin a new graph, and nodes are described within curly braces. A double-hyphen (--) is used to show relations between the nodes.
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The order- zero graph, K_0, is the unique graph having no vertices (hence its order is zero). It follows that K_0 also has no edges. Thus the null graph is a regular graph of degree zero. Some authors exclude K_0 from consideration as a graph (either by definition, or more simply as a matter of convenience).
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Any toroidal graph has chromatic number at most 7. The complete graph K7 provides an example of toroidal graph with chromatic number 7. Any triangle-free toroidal graph has chromatic number at most 4. By a result analogous to Fáry's theorem, any toroidal graph may be drawn with straight edges in a rectangle with periodic boundary conditions.
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It follows from the equivalent characterizations of trivially perfect graphs that every trivially perfect graph is also a cograph, a chordal graph, a Ptolemaic graph, an interval graph, and a perfect graph. The threshold graphs are exactly the graphs that are both themselves trivially perfect and the complements of trivially perfect graphs (co-trivially perfect graphs)., theorem 6.6.3, p.
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Petersen's graph and prisms. Moreover, 3-subdivision of any graph is 3-representable. In particular, for every graph G there exists a 3-representable graph H that contains G as a minor . A graph G is permutationally representable if it can be represented by a word of the form p1p2...pk, where pi is a permutation.
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A cycle double cover of the Petersen graph, corresponding to its embedding on the projective plane as a hemi-dodecahedron. In graph-theoretic mathematics, a cycle double cover is a collection of cycles in an undirected graph that together include each edge of the graph exactly twice. For instance, for any polyhedral graph, the faces of a convex polyhedron that represents the graph provide a double cover of the graph: each edge belongs to exactly two faces. It is an unsolved problem, posed by George Szekeres.
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In a directed plane graph, the dual graph may be made directed as well, by orienting each dual edge by a 90° clockwise turn from the corresponding primal edge.. Strictly speaking, this construction is not a duality of directed planar graphs, because starting from a graph and taking the dual twice does not return to itself, but instead constructs a graph isomorphic to the transpose graph of , the graph formed from by reversing all of its edges. Taking the dual four times returns to the original graph.
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Only five examples of vertex-transitive graph with no Hamiltonian cycles are known : the complete graph K2, the Petersen graph, the Coxeter graph and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle.Royle, G. "Cubic Symmetric Graphs (The Foster Census)." The characteristic polynomial of the Coxeter graph is (x-3) (x-2)^8 (x+1)^7 (x^2+2 x-1)^6. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
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In graph theory, the Henson graph is an undirected infinite graph, the unique countable homogeneous graph that does not contain an -vertex clique but that does contain all -free finite graphs as induced subgraphs. For instance, is a triangle-free graph that contains all finite triangle-free graphs. These graphs are named after C. Ward Henson, who published a construction for them (for all ) in 1971.. The first of these graphs, , is also called the homogeneous triangle-free graph or the universal triangle-free graph.
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In graph theory, a branch of mathematics, a periodic graph with respect to an operator F on graphs is one for which there exists an integer n > 0 such that Fn(G) is isomorphic to G. For example, every graph is periodic with respect to the complementation operator, whereas only complete graphs are periodic with respect to the operator that assigns to each graph the complete graph on the same vertices. Periodicity is one of many properties of graph operators, the central topic in graph dynamics.
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For a Hamiltonian decomposition to exist in an undirected graph, the graph must be connected and regular of even degree. A directed graph with such a decomposition must be strongly connected and all vertices must have the same in-degree and out-degree as each other, but this degree does not need to be even. The medial graph of the Herschel graph is a 4-regular planar graph with no Hamiltonian decomposition. The shaded regions correspond to the vertices of the underlying Herschel graph.
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A 1-factorization of a k-regular graph, a partition of the edges of the graph into perfect matchings, is the same thing as a k-edge- coloring of the graph. That is, a regular graph has a 1-factorization if and only if it is of class 1. As a special case of this, a 3-edge-coloring of a cubic (3-regular) graph is sometimes called a Tait coloring. Not every regular graph has a 1-factorization; for instance, the Petersen graph does not.
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The relative neighborhood graph is an example of a lens- based beta skeleton. It is a subgraph of the Delaunay triangulation. In turn, the Euclidean minimum spanning tree is a subgraph of it, from which it follows that it is a connected graph. The Urquhart graph, the graph formed by removing the longest edge from every triangle in the Delaunay triangulation, was originally proposed as a fast method to compute the relative neighborhood graph.. Although the Urquhart graph sometimes differs from the relative neighborhood graph.
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It may be interpreted as the clique complex of the comparability graph of the partial order. The matching complex of a graph consists of the sets of edges no two of which share an endpoint; again, this family of sets satisfies the no-Δ condition. It may be viewed as the clique complex of the complement graph of the line graph of the given graph. When the matching complex is referred to without any particular graph as context, it means the matching complex of a complete graph.
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In graph theory, a biconnected graph is a connected and "nonseparable" graph, meaning that if any one vertex were to be removed, the graph will remain connected. Therefore a biconnected graph has no articulation vertices. The property of being 2-connected is equivalent to biconnectivity, except that the complete graph of two vertices is usually not regarded as 2-connected. This property is especially useful in maintaining a graph with a two-fold redundancy, to prevent disconnection upon the removal of a single edge (or connection).
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Petersen graph as Kneser graph KG_{5,2} The Petersen graph is the complement of the line graph of K_5. It is also the Kneser graph KG_{5,2}; this means that it has one vertex for each 2-element subset of a 5-element set, and two vertices are connected by an edge if and only if the corresponding 2-element subsets are disjoint from each other. As a Kneser graph of the form KG_{2n-1,n-1} it is an example of an odd graph. Geometrically, the Petersen graph is the graph formed by the vertices and edges of the hemi-dodecahedron, that is, a dodecahedron with opposite points, lines and faces identified together.
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A factor-critical graph, together with perfect matchings of the subgraphs formed by removing one of its vertices. In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph.) is a graph with vertices in which every subgraph of vertices has a perfect matching. (A perfect matching in a graph is a subset of its edges with the property that each of its vertices is the endpoint of exactly one of the edges in the subset.) A matching that covers all but one vertex of a graph is called a near-perfect matching. So equivalently, a factor-critical graph is a graph in which there are near-perfect matchings that avoid every possible vertex.
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An outerplanar graph is an undirected graph that can be drawn in the plane without crossings in such a way that all of the vertices belong to the unbounded face of the drawing. That is, no vertex is totally surrounded by edges. Alternatively, a graph G is outerplanar if the graph formed from G by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph.. A maximal outerplanar graph is an outerplanar graph that cannot have any additional edges added to it while preserving outerplanarity. Every maximal outerplanar graph with n vertices has exactly 2n − 3 edges, and every bounded face of a maximal outerplanar graph is a triangle.
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Thus the graph of can be obtained from the graph of by switching the positions of the and axes. This is equivalent to reflecting the graph across the line .
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In the utility graph, and , violating this inequality, so the utility graph cannot be planar..
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A graph that is itself connected has exactly one component, consisting of the whole graph.
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An alternative construction, the medial graph, coincides with the line graph for planar graphs with maximum degree three, but is always planar. It has the same vertices as the line graph, but potentially fewer edges: two vertices of the medial graph are adjacent if and only if the corresponding two edges are consecutive on some face of the planar embedding. The medial graph of the dual graph of a plane graph is the same as the medial graph of the original plane graph.. For regular polyhedra or simple polyhedra, the medial graph operation can be represented geometrically by the operation of cutting off each vertex of the polyhedron by a plane through the midpoints of all its incident edges.. This operation is known variously as the second truncation,. degenerate truncation,.
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An Eulerian graph G (a connected graph in which every vertex has even degree) necessarily has an Euler tour, a closed walk passing through each edge of G exactly once. This tour corresponds to a Hamiltonian cycle in the line graph L(G), so the line graph of every Eulerian graph is Hamiltonian. Line graphs may have other Hamiltonian cycles that do not correspond to Euler tours, and in particular the line graph L(G) of every Hamiltonian graph G is itself Hamiltonian, regardless of whether the graph G is Eulerian.. A tournament (with more than two vertices) is Hamiltonian if and only if it is strongly connected. The number of different Hamiltonian cycles in a complete undirected graph on n vertices is and in a complete directed graph on n vertices is .
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In the mathematical field of graph theory, the Clebsch graph is either of two complementary graphs on 16 vertices, a 5-regular graph with 40 edges and a 10-regular graph with 80 edges. The 80-edge variant is the order-5 halved cube graph; it was called the Clebsch graph name by Seidel (1968)J. J. Seidel, Strongly regular graphs with (−1,1,0) adjacency matrix having eigenvalue 3, Lin. Alg. Appl. 1 (1968) 281-298.
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The primal constraint graph or simply primal graph (also the Gaifman graph) of a constraint satisfaction problem is the graph whose nodes are the variables of the problem and an edge joins a pair of variables if the two variables occur together in a constraint.Handbook of Constraint Programming, by Francesca Rossi, Peter Van Beek, Toby Walsh (2006) , p. 211, 212 The primal constraint graph is in fact the primal graph of the constraint hypergraph.
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A representation of a chordal graph as an intersection of subtrees forms a tree decomposition of the graph, with treewidth equal to one less than the size of the largest clique in the graph; the tree decomposition of any graph G can be viewed in this way as a representation of G as a subgraph of a chordal graph. The tree decomposition of a graph is also the junction tree of the junction tree algorithm.
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A minor of a graph G is any graph H that is isomorphic to a graph that can be obtained from a subgraph of G by contracting some edges. If G does not have a graph H as a minor, then we say that G is H-free. Let H be a fixed graph. Intuitively, if G is a huge H-free graph, then there ought to be a "good reason" for this.
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The block graph of a given graph G is the intersection graph of its blocks. Thus, it has one vertex for each block of G, and an edge between two vertices whenever the corresponding two blocks share a vertex. A graph H is the block graph of another graph G exactly when all the blocks of H are complete subgraphs. The graphs H with this property are known as the block graphs..
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Every matchstick graph is a unit distance graph. Penny graphs are the graphs that can be represented by tangencies of non-overlapping unit circles. Every penny graph is a matchstick graph. However, some matchstick graphs (such as the eight-vertex cubic matchstick graph of the first illustration) are not penny graphs, because realizing them as a matchstick graph causes some non-adjacent vertices to be closer than unit distance to each other.
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The complete graph K6, the Petersen graph, and the other five graphs in the Petersen family do not have linkless embeddings. Every graph minor of a linklessly embeddable graph is again linklessly embeddable, as is every graph that can be reached from a linklessly embeddable graph by a Y-Δ transform. The linklessly embeddable graphs have the Petersen family graphs as their forbidden minors,. and include the planar graphs and apex graphs.
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A 3-outerplanar graph, the graph of a rhombic dodecahedron. There are four vertices on the outside face, eight vertices on the second layer (light yellow), and two vertices on the third layer (darker yellow). Because of the symmetries of the graph, no other embedding has fewer layers. In graph theory, a k-outerplanar graph is a planar graph that has a planar embedding in which the vertices belong to at most k concentric layers.
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According to the Foster census, the Nauru graph is the only cubic symmetric graph on 24 vertices. The generalized Petersen graph G(n,k) is vertex-transitive if and only if n = 10 and k =2 or if k2 ≡ ±1 (mod n) and is edge-transitive only in the following seven cases: (n,k) = (4,1), (5,2), (8,3), (10,2), (10,3), (12,5), (24,5).. So the Nauru graph is one of only seven symmetric Generalized Petersen graphs. Among these seven graphs are the cubical graph G(4,1), the Petersen graph G(5,2), the Möbius–Kantor graph G(8,3), the dodecahedral graph G(10,2) and the Desargues graph G(10,3). The Nauru graph is a Cayley graph of S4, the symmetric group of permutations on four elements, generated by the three different ways of swapping the first element with one of the three others : (1 2), (1 3) and (1 4).
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An example of a map in graph theory In graph theory, a map is a drawing of a graph on a surface without overlapping edges (an embedding). If the surface is a plane then a map is a planar graph, similar to a political map.
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This construction of the complement graph is an instance of the same process of choosing randomly and independently whether to include each edge, so it also (with probability 1) generates the Rado graph. Therefore, the Rado graph is a self-complementary graph., Proposition 5.
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One of them relates the coefficients of the characteristic polynomial of a graph to certain structural features of the graph. Another one is a simple relation between the characteristic polynomials of a graph and its line graph. Sachs subgraphs are also named after Sachs.
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A graph vertex coloring is a weak coloring, but not necessarily vice versa. Every graph has a weak 2-coloring. The figure on the right illustrates a simple algorithm for constructing a weak 2-coloring in an arbitrary graph. Part (a) shows the original graph.
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In graph theory, a clique graph of an undirected graph G is another graph K(G) that represents the structure of cliques in G. Clique graphs were discussed at least as early as 1968, and a characterization of clique graphs was given in 1971.
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A polytree (or directed tree or oriented tree or singly connected network) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest.
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In the mathematical fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of the numbers of matchings of various sizes in a graph. It is one of several graph polynomials studied in algebraic graph theory.
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Australasian J. Combin. 69 (2017), 105−118. it is shown that any 132-representable graph is necessarily a circle graph, and any tree and any cycle graph, as well as any graph on at most 5 vertices, are 132-representable. It was shown in Mandelshtam.
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Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976. Moreover, the Harries–Wong graph and Harries graph are cospectral graphs.
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Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976. Moreover, the Harries-Wong graph and Harries graph are cospectral graphs.
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In the mathematical area of graph theory, a contact graph or tangency graph is a graph whose vertices are represented by geometric objects (e.g. curves, line segments, or polygons), and whose edges correspond to two objects touching (but not crossing) according to some specified notion. online PDF It is similar to the notion of an intersection graph but differs from it in restricting the ways that the underlying objects are allowed to intersect each other. The circle packing theorem states that every planar graph can be represented as a contact graph of circles.
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In the mathematical field of graph theory, the Desargues graph is a distance- transitive cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises from several different combinatorial constructions, has a high level of symmetry, is the only known non-planar cubic partial cube, and has been applied in chemical databases. The name "Desargues graph" has also been used to refer to a ten-vertex graph, the complement of the Petersen graph, which can also be formed as the bipartite half of the 20-vertex Desargues graph..
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With the 3\times 3 Rook's graph and the Brouwer–Haemers graph, the Games graph is one of only three possible strongly regular graphs whose parameters have the form \bigl((n^2+3n-1)^2,n^2(n+3),1,n(n+1)\bigr). The same properties that produce a strongly regular graph from a cap set can also be used with an 11-point cap set in PG(4,3), producing a smaller strongly regular graph with parameters (243,22,1,2). This graph is the Berlekamp–van Lint–Seidel graph.
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The eight 6-vertex asymmetric graphs The Frucht graph, one of the five smallest asymmetric cubic graphs. In graph theory, a branch of mathematics, an undirected graph is called an asymmetric graph if it has no nontrivial symmetries. Formally, an automorphism of a graph is a permutation p of its vertices with the property that any two vertices u and v are adjacent if and only if p(u) and p(v) are adjacent. The identity mapping of a graph onto itself is always an automorphism, and is called the trivial automorphism of the graph.
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The Möbius–Kantor graph is a subgraph of the four-dimensional hypercube graph, formed by removing eight edges from the hypercube . Since the hypercube is a unit distance graph, the Möbius–Kantor graph can also be drawn in the plane with all edges unit length, although such a drawing will necessarily have some pairs of crossing edges. The Möbius–Kantor graph also occurs many times as in induced subgraph of the Hoffman–Singleton graph. Each of these instances is in fact an eigenvector of the Hoffman-Singleton graph, with associated eigenvalue -3.
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The Petersen family. As showed, each of the seven graphs of the Petersen family is intrinsically linked: no matter how each of these graphs is embedded in space, they have two cycles that are linked to each other. These graphs include the complete graph K6, the Petersen graph, the graph formed by removing an edge from the complete bipartite graph K4,4, and the complete tripartite graph K3,3,1. Every planar graph has a flat and linkless embedding: simply embed the graph into a plane and embed the plane into space.
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A strangulated graph, formed by using clique-sums to glue together a maximal planar graph (yellow) and two chordal graphs (red and blue). The red chordal graph can in turn be decomposed into clique-sums of four maximal planar graphs (two edges and two triangles). In graph theoretic mathematics, a strangulated graph is a graph in which deleting the edges of any induced cycle of length greater than three would disconnect the remaining graph. That is, they are the graphs in which every peripheral cycle is a triangle.
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Additionally, every well-covered graph is a critical graph for vertex covering in the sense that, for every vertex , deleting from the graph produces a graph with a smaller minimum vertex cover. The independence complex of a graph is the simplicial complex having a simplex for each independent set in . A simplicial complex is said to be pure if all its maximal simplices have the same cardinality, so a well-covered graph is equivalently a graph with a pure independence complex.For examples of papers using this terminology, see and .
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External memory graph traversal is a type of graph traversal optimized for accessing externally stored memory.
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When Π is a cyclic group, the voltage graph may be called a cyclic-voltage graph.
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A graph with connectivity 4. In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed. The vertex- connectivity, or just connectivity, of a graph is the largest k for which the graph is k-vertex-connected.
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An infinite-fold abelian covering graph of a finite (multi)graph is called a topological crystal, an abstraction of crystal structures. For example, the diamond crystal as a graph is the maximal abelian covering graph of the four-edge dipole graph. This view combined with the idea of "standard realizations" turns out to be useful in a systematic design of (hypothetical) crystals.
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It is NP-hard, and more specifically complete for the existential theory of the reals, to test whether a given graph is a unit distance graph, or is a strict unit distance graph. It is also NP-complete to determine whether a unit distance graph has a Hamiltonian cycle, even when the vertices of the graph all have integer coordinates.
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Graph databases are technologies that are translations of the relational online transaction processing (OLTP) databases. On the other hand, graph compute engines are used in online analytical processing (OLAP) for bulk analysis. Graph databases attracted considerable attention in the 2000s, due to the successes of major technology corporations in using proprietary graph databases, along with the introduction of open- source graph databases.
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Sometimes graph grammar is used as a synonym for graph rewriting system, especially in the context of formal languages; the different wording is used to emphasize the goal of constructions, like the enumeration of all graphs from some starting graph, i.e. the generation of a graph language – instead of simply transforming a given state (host graph) into a new state.
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In graph theory, the Shannon capacity of a graph is a graph invariant defined from the number of independent sets of strong graph products. It measures the Shannon capacity of a communications channel defined from the graph, and is upper bounded by the Lovász number, which can be computed in polynomial time. However, the computational complexity of the Shannon capacity itself remains unknown.
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A circle with five chords and the corresponding circle graph. In graph theory, a circle graph is the intersection graph of a set of chords of a circle. That is, it is an undirected graph whose vertices can be associated with chords of a circle such that two vertices are adjacent if and only if the corresponding chords cross each other.
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In the mathematical field of graph theory, the Meringer graph is a 5-regular undirected graph with 30 vertices and 75 edges named after Markus Meringer.. It is one of the four (5,5)-cage graphs, the others being the Foster cage, the Robertson–Wegner graph, and the Wong graph. It has chromatic number 3, diameter 3, and is 5-vertex-connected.
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Tietze's graph is isomorphic to the graph J3, part of an infinite family of flower snarks introduced by R. Isaacs in 1975.. Unlike the Petersen graph, the Tietze graph can be covered by four perfect matchings. This property plays a key role in a proof that testing whether a graph can be covered by four perfect matchings is NP-complete..
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In graph theory, the Games graph is the largest known locally linear strongly regular graph. Its parameters as a strongly regular graph are (729,112,1,20). This means that it has 729 vertices, and 40824 edges (112 per vertex). Each edge is in a unique triangle (it is a locally linear graph) and each non- adjacent pair of vertices have exactly 20 shared neighbors.
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Conder, Malnič, Marušič, Pisanski and Potočnik rediscovered this 112-vertices graph in 2002 and named it the Ljubljana graph after the capital of Slovenia.Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; and Potočnik, P. "The Ljubljana Graph." 2002. . They proved that it was the unique 112-vertices edge- but not vertex-transitive cubic graph and therefore that was the graph found by Foster.
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Seven intervals on the real line and the corresponding seven-vertex interval graph. In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals. Interval graphs are chordal graphs and perfect graphs.
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In the mathematical field of graph theory, the Franklin graph is a 3-regular graph with 12 vertices and 18 edges. The Franklin graph is named after Philip Franklin, who disproved the Heawood conjecture on the number of colors needed when a two-dimensional surface is partitioned into cells by a graph embedding.Franklin, P. "A Six Color Problem." J. Math. Phys.
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The utility graph K3,3 is a circulant graph. It is the (3,4)-cage, the smallest triangle-free cubic graph. Like all other complete bipartite graphs, it is a well-covered graph, meaning that every maximal independent set has the same size. In this graph, the only two maximal independent sets are the two sides of the bipartition, and obviously they are equal.
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In the following example, graph G and graph H are homeomorphic. If G' is the graph created by subdivision of the outer edges of G and H' is the graph created by subdivision of the inner edge of H, then G' and H' have a similar graph drawing: Therefore, there exists an isomorphism between G' and H', meaning G and H are homeomorphic.
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In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e.
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The Rado graph was first constructed by in two ways, with vertices either the hereditarily finite sets or the natural numbers. (Strictly speaking Ackermann described a directed graph, and the Rado graph is the corresponding undirected graph given by forgetting the directions on the edges.) constructed the Rado graph as the random graph on a countable number of points. They proved that it has infinitely many automorphisms, and their argument also shows that it is unique though they did not mention this explicitly. rediscovered the Rado graph as a universal graph, and gave an explicit construction of it with vertex set the natural numbers.
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Representation of a planar graph as a string graph. Every planar graph is a string graph: credit this observation to . one may form a string graph representation of an arbitrary plane-embedded graph by drawing a string for each vertex that loops around the vertex and around the midpoint of each adjacent edge, as shown in the figure. For any edge uv of the graph, the strings for u and v cross each other twice near the midpoint of uv, and there are no other crossings, so the pairs of strings that cross represent exactly the adjacent pairs of vertices of the original planar graph.
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In the mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage or Cremona–Richmond graph is a 3-regular graph with 30 vertices and 45 edges. As the unique smallest cubic graph of girth 8 it is a cage and a Moore graph. It is bipartite, and can be constructed as the Levi graph of the generalized quadrangle W2 (known as the Cremona–Richmond configuration). The graph is named after William Thomas Tutte and H. S. M. Coxeter; it was discovered by Tutte (1947) but its connection to geometric configurations was investigated by both authors in a pair of jointly published papers (Tutte 1958; Coxeter 1958a).
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In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one. Spectral graph theory is also concerned with graph parameters that are defined via multiplicities of eigenvalues of matrices associated to the graph, such as the Colin de Verdière number.
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In mathematics, a biased graph is a graph with a list of distinguished circles (edge sets of simple cycles), such that if two circles in the list are contained in a theta graph, then the third circle of the theta graph is also in the list. A biased graph is a generalization of the combinatorial essentials of a gain graph and in particular of a signed graph. Formally, a biased graph Ω is a pair (G, B) where B is a linear class of circles; this by definition is a class of circles that satisfies the theta-graph property mentioned above. A subgraph or edge set whose circles are all in B (and which contains no half-edges) is called balanced.
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A graph is 5-vertex-connected when there are no five vertices whose removal leaves a disconnected graph. The complete graph is a graph with an edge between every five vertices, and a subdivision of a complete graph modifies this by replacing some of its edges by longer paths. So a graph contains a subdivision of if it is possible to pick out five vertices of , and a set of ten paths connecting these five vertices in pairs without any of the paths sharing vertices or edges with each other. In any drawing of the graph on the Euclidean plane, at least two of the ten paths must cross each other, so a graph G that contains a K5 subdivision cannot be a planar graph.
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A graph is said to be -factor-critical if every subset of vertices has a perfect matching. Under this definition, a hypomatchable graph is 1-factor-critical.. Even more generally, a graph is -factor-critical if every subset of vertices has an -factor, that is, it is the vertex set of an -regular subgraph of the given graph. A critical graph (without qualification) is usually assumed to mean a graph for which removing each of its vertices reduces the number of colors it needs in a graph coloring. The concept of criticality has been used much more generally in graph theory to refer to graphs for which removing each possible vertex changes or does not change some relevant property of the graph.
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In graph theory, a tolerance graph is an undirected graph in which every vertex can be represented by a closed interval and a real number called its tolerance, in such a way that two vertices are adjacent in the graph whenever their intervals overlap in a length that is at least the minimum of their two tolerances. This class of graphs was introduced in 1982 by Martin Charles Golumbic and Clyde Monma, who used them to model scheduling problems in which the tasks to be modeled can share resources for limited amounts of time. Every interval graph is a tolerance graph. The complement graph of every tolerance graph is a perfectly orderable graph, from which it follows that the tolerance graphs themselves are perfect graphs.
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However, it is NP-complete to find the largest Halin subgraph of a given graph, to test whether there exists a Halin subgraph that includes all vertices of a given graph, or to test whether a given graph is a subgraph of a larger Halin graph..
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In computer science, graph traversal (also known as graph search) refers to the process of visiting (checking and/or updating) each vertex in a graph. Such traversals are classified by the order in which the vertices are visited. Tree traversal is a special case of graph traversal.
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In graph theory, precoloring extension is the problem of extending a graph coloring of a subset of the vertices of a graph, with a given set of colors, to a coloring of the whole graph that does not assign the same color to any two adjacent vertices.
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The Wiener–Araya graph is, in graph theory, a graph on 42 vertices with 67 edges. It is hypohamiltonian, which means that it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It is also planar.
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In the mathematical field of graph theory, the Robertson graph or (4,5)-cage, is a 4-regular undirected graph with 19 vertices and 38 edges named after Neil Robertson.Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.
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Outerplanar graphs have a forbidden graph characterization analogous to Kuratowski's theorem and Wagner's theorem for planar graphs: a graph is outerplanar if and only if it does not contain a subdivision of the complete graph K4 or the complete bipartite graph K2,3.; ; , Proposition 7.3.1, p. 117; .
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The corresponding concept for undirected graphs is a forest, an undirected graph without cycles. Choosing an orientation for a forest produces a special kind of directed acyclic graph called a polytree. However, there are many other kinds of directed acyclic graph that are not formed by orienting the edges of an undirected acyclic graph. Moreover, every undirected graph has an acyclic orientation, an assignment of a direction for its edges that makes it into a directed acyclic graph.
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Two different power graphs that represent the same graph. In general, there is no unique minimal power graph for a given graph. In this example (right) a graph of four nodes and five edges admits two minimal power graphs of two power edges each. The main difference between these two minimal power graphs is the higher nesting level of the second power graph as well as a loss of symmetry with respect to the underlying graph.
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The automorphism group of the Coxeter graph is a group of order 336.Royle, G. F028A data It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Coxeter graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the Coxeter graph, referenced as F28A, is the only cubic symmetric graph on 28 vertices.
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Including the Heawood graph, there are 8 distinct graphs of order 14 with crossing number 3. The Heawood graph is the smallest cubic graph with Colin de Verdière graph invariant μ = 6. The Heawood graph is a unit distance graph: it can be embedded in the plane such that adjacent vertices are exactly at distance one apart, with no two vertices embedded to the same point and no vertex embedded into a point within an edge..
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A forest is an undirected graph in which any two vertices are connected by at most one path. Equivalently, a forest is an undirected acyclic graph. Equivalently, a forest is an undirected graph, all of whose connected components are trees; in other words, the graph consists of a disjoint union of trees. As special cases, the order-zero graph (a forest consisting of zero trees), a single tree, and an edgeless graph, are examples of forests.
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A cycle graph or circular graph of order is a graph in which the vertices can be listed in an order v1, v2, …, vn such that the edges are the where i = 1, 2, …, n − 1, plus the edge . Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph.
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A plane graph (in blue) and its medial graph (in red). In the mathematical discipline of graph theory, the medial graph of plane graph G is another graph M(G) that represents the adjacencies between edges in the faces of G. Medial graphs were introduced in 1922 by Ernst Steinitz to study combinatorial properties of convex polyhedra, although the inverse construction was already used by Peter Tait in 1877 in his foundational study of knots and links.
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In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with n vertices is called Cn. The number of vertices in Cn equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it.
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The Errera graph, on the other hand, provides a counterexample to Kempe's entire method. When this method is run on the Errera graph, starting with no vertices colored, it can fail to find a valid coloring for the whole graph. Additionally, unlike the Poussin graph, all vertices in the Errera graph have degree five or more. Therefore, on this graph, it is impossible to avoid the problematic cases of Kempe's method by choosing lower-degree vertices.
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The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties.
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The chromatic number of the graph is exactly the minimum makespan, the optimal time to finish all jobs without conflicts. Details of the scheduling problem define the structure of the graph. For example, when assigning aircraft to flights, the resulting conflict graph is an interval graph, so the coloring problem can be solved efficiently. In bandwidth allocation to radio stations, the resulting conflict graph is a unit disk graph, so the coloring problem is 3-approximable.
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It contains as an induced subgraph the Grötzsch graph, the smallest triangle-free four-chromatic graph, and every four-chromatic induced subgraph of the Clebsch graph is a supergraph of the Grötzsch graph. More strongly, every triangle-free four-chromatic graph with no induced path of length six or more is an induced subgraph of the Clebsch graph and an induced supergraph of the Grötzsch graph.. The 5-regular Clebsch graph is the Keller graph of dimension two, part of a family of graphs used to find tilings of high- dimensional Euclidean spaces by hypercubes no two of which meet face-to-face. The 5-regular Clebsch graph can be embedded as a regular map in the orientable manifold of genus 5, forming pentagonal faces; and in the non-orientable surface of genus 6, forming tetragonal faces.
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A dot product representation of a simple graph is a method of representing a graph using vector spaces and the dot product from linear algebra. Every graph has a dot product representation....
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Each vertex not in the induced Möbius–Kantor graph is adjacent to exactly four vertices in the Möbius–Kantor graph, two each in half of a bipartition of the Möbius–Kantor graph.
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Microsoft Automatic Graph Layout (MSAGL) is a .NET library for automatic graph layout. It was created by Lev Nachmanson at Microsoft Research. Earlier versions carried the name GLEE (Graph Layout Execution Engine).
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In the analysis of algorithms on graphs, the distinction between a graph and its complement is an important one, because a sparse graph (one with a small number of edges compared to the number of pairs of vertices) will in general not have a sparse complement, and so an algorithm that takes time proportional to the number of edges on a given graph may take a much larger amount of time if the same algorithm is run on an explicit representation of the complement graph. Therefore, researchers have studied algorithms that perform standard graph computations on the complement of an input graph, using an implicit graph representation that does not require the explicit construction of the complement graph. In particular, it is possible to simulate either depth-first search or breadth-first search on the complement graph, in an amount of time that is linear in the size of the given graph, even when the complement graph may have a much larger size. It is also possible to use these simulations to compute other properties concerning the connectivity of the complement graph...
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Therefore, a chordal graph is also moral. But a moral graph is not necessarily chordal., p. 50.
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One particularly well- known class of closed graph theorems are the closed graph theorems in functional analysis.
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In graph theory, an st-planar graph is a bipolar orientation of a plane graph for which both the source and the sink of the orientation are on the outer face of the graph. That is, it is a directed graph drawn without crossings in the plane, in such a way that there are no directed cycles in the graph, exactly one graph vertex has no incoming edges, exactly one graph vertex has no outgoing edges, and these two special vertices both lie on the outer face of the graph.. Within the drawing, each face of the graph must have the same structure: there is one vertex that acts as the source of the face, one vertex that acts as the sink of the face, and all edges within the face are directed along two paths from the source to the sink. If one draws an additional edge from the sink of an st-planar graph back to the source, through the outer face, and then constructs the dual graph (oriented each dual edge clockwise with respect to its primal edge) then the result is again an st-planar graph, augmented with an extra edge in the same way.
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In graph theory, a branch of mathematics, graph canonization is the problem finding a canonical form of a given graph G. A canonical form is a labeled graph Canon(G) that is isomorphic to G, such that every graph that is isomorphic to G has the same canonical form as G. Thus, from a solution to the graph canonization problem, one could also solve the problem of graph isomorphism: to test whether two graphs G and H are isomorphic, compute their canonical forms Canon(G) and Canon(H), and test whether these two canonical forms are identical. The canonical form of a graph is an example of a complete graph invariant: every two isomorphic graphs have the same canonical form, and every two non-isomorphic graphs have different canonical forms... Conversely, every complete invariant of graphs may be used to construct a canonical form.. The vertex set of an n-vertex graph may be identified with the integers from 1 to n, and using such an identification a canonical form of a graph may also be described as a permutation of its vertices. Canonical forms of a graph are also called canonical labelings,. and graph canonization is also sometimes known as graph canonicalization.
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Let and be two graphs. Graph is a sub-graph of graph (written as ) if and . If and contains all of the edges with , then is an induced sub- graph of . We call and isomorphic (written as ), if there exists a bijection (one-to-one correspondence) with for all .
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The half graph has a unique perfect matching. This is straightforward to see by induction: u_n must be matched to its only neighbor, v_n, and the remaining vertices form another half graph. More strongly, every bipartite graph with a unique perfect matching is a subgraph of a half graph.
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Similarly, in the context of graph theory, if the number of links is close to its maximum, then the graph would be known as dense graph. If the number of links is lower than the maximum number of links, this type of graphs are referred as sparse graph.
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A finite graph is a graph in which the vertex set and the edge set are finite sets. Otherwise, it is called an infinite graph. Most commonly in graph theory it is implied that the graphs discussed are finite. If the graphs are infinite, that is usually specifically stated.
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As observes, every comparability graph that is neither complete nor bipartite has a skew partition. The complement of any interval graph is a comparability graph. The comparability relation is called an interval order. Interval graphs are exactly the graphs that are chordal and that have comparability graph complements.
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The Gremlin graph traversal machine can execute on a single machine or across a multi-machine compute cluster. Execution agnosticism allows Gremlin to run over both graph databases (OLTP) and graph processors (OLAP).
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The spectrum of a McKay–Miller–Širáň graph has at most five distinct eigenvalues. In some of these graphs, all of these values are integers, so that the graph is an integral graph.
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The F26A graph can be embedded as a chiral regular map in the torus, with 13 hexagonal faces. The dual graph for this embedding is isomorphic to the Paley graph of order 13.
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A form of Wagner's theorem applies for bipartite minors: A bipartite graph G is a planar graph if and only if it does not have the utility graph K3,3 as a bipartite minor..
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For partial 3-trees there are four forbidden minors: the complete graph on five vertices, the octahedral graph with six vertices, the eight-vertex Wagner graph, and the pentagonal prism with ten vertices..
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A minor of a graph G is another graph formed from G by contracting edges, removing edges, and removing vertices. Graph minors have a deep theory in which several important results involve pathwidth.
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The vertices of the Cayley graph are the inverse permutations of those in the permutohedron.This Cayley graph labeling is shown, e.g., by . The image on the right shows the Cayley graph of S4.
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In the mathematical discipline of graph theory, the (m,n)-lollipop graph is a special type of graph consisting of a complete graph (clique) on m vertices and a path graph on n vertices, connected with a bridge. The special case of the (2n/3,n/3)-lollipop graphs are known as graphs which achieve the maximum possible hitting time, cover time and commute time.
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A drawing of the Petersen graph with slope number 3 In graph drawing and geometric graph theory, the slope number of a graph is the minimum possible number of distinct slopes of edges in a drawing of the graph in which vertices are represented as points in the Euclidean plane and edges are represented as line segments that do not pass through any non-incident vertex.
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The same is true for detecting whether the pattern graph is an induced subgraph of the larger graph, or whether it has a graph homomorphism to the larger graph., Corollary 18.1, p. 401.. For the same reason, the problem of testing whether a graph of bounded book thickness obeys a given formula of first order logic is fixed-parameter tractable., Theorem 18.7, p. 405.
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In the mathematical field of graph theory, the Harries-Wong graph is a 3-regular undirected graph with 70 vertices and 105 edges. The Harries-Wong graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 and is Hamiltonian. It is also a 3-vertex-connected and 3-edge-connected non-planar cubic graph. It has book thickness 3 and queue number 2.
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The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. For the existence of Eulerian trails it is necessary that zero or two vertices have an odd degree; this means the Königsberg graph is not Eulerian.
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It is the smallest vertex-transitive graph that is not a Cayley graph.As stated, this assumes that Cayley graphs need not be connected. Some sources require Cayley graphs to be connected, making the two- vertex empty graph the smallest vertex-transitive non-Cayley graph; under the definition given by these sources, the Petersen graph is the smallest connected vertex-transitive graph that is not Cayley.
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An empty graph is a graph that has an empty set of vertices (and thus an empty set of edges). The order of a graph is its number of vertices . The size of a graph is its number of edges . However, in some contexts, such as for expressing the computational complexity of algorithms, the size is (otherwise, a non-empty graph could have a size 0).
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Berge conjectured the converse, that every Berge graph is perfect. This was finally proven as the strong perfect graph theorem of Chudnovsky, Robertson, Seymour, and Thomas (2006). It trivially implies the perfect graph theorem, hence the name. The perfect graph theorem has a short proof, but the proof of the strong perfect graph theorem is long and technical, based on a deep structural decomposition of Berge graphs.
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The Laves graph In geometry and crystallography, the Laves graph is an infinite cubic symmetric graph. It can be embedded into three-dimensional space, with integer coordinates, to form a structure with chiral symmetry. . in which the three edges at each vertex form 120° angles to each other. It can also be defined more abstractly as a covering graph of the complete graph on four vertices.
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The Dürer graph G(6, 2). In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter.
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In the mathematical area of graph theory, the Mycielskian or Mycielski graph of an undirected graph is a larger graph formed from it by a construction of . The construction preserves the property of being triangle-free but increases the chromatic number; by applying the construction repeatedly to a triangle- free starting graph, Mycielski showed that there exist triangle-free graphs with arbitrarily large chromatic number.
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In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph.
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The butterfly graph has diameter 2 and girth 3, radius 1, chromatic number 3, chromatic index 4 and is both Eulerian and a penny graph (this implies that it is unit distance and planar). It is also a 1-vertex- connected graph and a 2-edge-connected graph. There are only 3 non-graceful simple graphs with five vertices. One of them is the butterfly graph.
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In computer science, a linear graph grammar (also a connection graph reduction system or a port graph grammarBawden (1986) introduces the formalism calling them connection graphs. ) is a class of graph grammar on which nodes have a number of ports connected together by edges and edges connect exactly two ports together. Interaction nets are a special subclass of linear graph grammars in which rewriting is confluent.
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The Johnson graph is the graph whose vertices are the -element subsets of an -element set, two vertices being adjacent when they meet in a -element set. The Johnson graph is the complement of the Kneser graph . Johnson graphs are closely related to the Johnson scheme, both of which are named after Selmer M. Johnson. The generalized Kneser graph has the same vertex set as the Kneser graph , but connects two vertices whenever they correspond to sets that intersect in or fewer items .
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In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. The Laplacian matrix can be used to find many useful properties of a graph. Together with Kirchhoff's theorem, it can be used to calculate the number of spanning trees for a given graph. The sparsest cut of a graph can be approximated through the second smallest eigenvalue of its Laplacian by Cheeger's inequality.
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In graph theory, a string graph is an intersection graph of curves in the plane; each curve is called a "string". Given a graph G, G is a string graph if and only if there exists a set of curves, or strings, drawn in the plane such that no three strings intersect at a single point and such that the graph having a vertex for each curve and an edge for each intersecting pair of curves is isomorphic to G.
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The 16-vertex Möbius ladder, an example of a nearly planar graph. If a graph is an apex graph, it is not necessarily the case that it has a unique apex. For instance, in the minor- minimal nonplanar graphs K5 and K3,3, any of the vertices can be chosen as the apex. defined a nearly planar graph to be a nonplanar apex graph with the property that all vertices can be the apex of the graph; thus, K5 and K3,3 are nearly planar.
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A path graph or linear graph of order is a graph in which the vertices can be listed in an order v1, v2, …, vn such that the edges are the where i = 1, 2, …, n − 1. Path graphs can be characterized as connected graphs in which the degree of all but two vertices is 2 and the degree of the two remaining vertices is 1. If a path graph occurs as a subgraph of another graph, it is a path in that graph.
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In the mathematical field of graph theory, the Barnette–Bosák–Lederberg graph is a cubic (that is, 3-regular) polyhedral graph with no Hamiltonian cycle, the smallest such graph possible. It was discovered in the mid-1960s by Joshua Lederberg, David Barnette, and Juraj Bosák, after whom it is named. It has 38 vertices and 69 edges. Other larger non-Hamiltonian cubic polyhedral graphs include the 46-vertex Tutte graph and a 44-vertex graph found by Emanuels Grīnbergs using Grinberg's theorem.
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In the mathematical field of graph theory, a half-transitive graph is a graph that is both vertex-transitive and edge-transitive, but not symmetric. In other words, a graph is half-transitive if its automorphism group acts transitively upon both its vertices and its edges, but not on ordered pairs of linked vertices. The Holt graph is the smallest half-transitive graph. The lack of reflectional symmetry in this drawing highlights the fact that edges are not equivalent to their inverse.
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An accessible pointed graph is a directed graph with a distinguished vertex (the "root") such that for any node in the graph there is at least one path in the directed graph from the root to that node. The anti-foundation axiom postulates that each such directed graph corresponds to the membership structure of a unique set. For example, the directed graph with only one node and an edge from that node to itself corresponds to a set of the form x = {x}.
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In chemical graph theory and in mathematical chemistry, a molecular graph or chemical graph is a representation of the structural formula of a chemical compound in terms of graph theory. A chemical graph is a labeled graph whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds. Its vertices are labeled with the kinds of the corresponding atoms and edges are labeled with the types of bonds. For particular purposes any of the labelings may be ignored.
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In graph theory, a branch of mathematics, a -biclique-free graph is a graph that has no 2-vertex complete bipartite graph as a subgraph. A family of graphs is biclique-free if there exists a number such that the graphs in the family are all -biclique-free. The biclique-free graph families form one of the most general types of sparse graph family. They arise in incidence problems in discrete geometry, and have also been used in parameterized complexity.
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If there are vertices in the graph, then each spanning tree has edges. This figure shows there may be more than one minimum spanning tree in a graph. In the figure, the two trees below the graph are two possibilities of minimum spanning tree of the given graph. There may be several minimum spanning trees of the same weight; in particular, if all the edge weights of a given graph are the same, then every spanning tree of that graph is minimum.
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A comparability graph is an undirected graph formed from a partial order by creating a vertex per element of the order, and an edge connecting any two comparable elements. Thus, a clique in a comparability graph corresponds to a chain, and an independent set in a comparability graph corresponds to an antichain. Any induced subgraph of a comparability graph is itself a comparability graph, formed from the restriction of the partial order to a subset of its elements. An undirected graph is perfect if, in every induced subgraph, the chromatic number equals the size of the largest clique.
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A cluster graph with clusters (complete subgraphs) of sizes 1, 2, 3, 4, 4, 5, and 6 In graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs. Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster graphs are also called P3-free graphs. They are the complement graphs of the complete multipartite graphsCluster graphs, Information System on Graph Classes and their Inclusions, accessed 2016-06-26. and the 2-leaf powers..
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In graph theory, a branch of mathematics, graph canonization is the problem of finding a canonical form of a given graph G. A canonical form is a labeled graph Canon(G) that is isomorphic to G, such that every graph that is isomorphic to G has the same canonical form as G. Thus, from a solution to the graph canonization problem, one could also solve the problem of graph isomorphism: to test whether two graphs G and H are isomorphic, compute their canonical forms Canon(G) and Canon(H), and test whether these two canonical forms are identical.
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Applying the back-and-forth construction to any two isomorphic finite subgraphs of the Rado graph extends their isomorphism to an automorphism of the entire Rado graph. The fact that every isomorphism of finite subgraphs extends to an automorphism of the whole graph is expressed by saying that the Rado graph is ultrahomogeneous. In particular, there is an automorphism taking any ordered pair of adjacent vertices to any other such ordered pair, so the Rado graph is a symmetric graph. The automorphism group of the Rado graph is a simple group, whose number of elements is the cardinality of the continuum.
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Although the Rado graph is universal for induced subgraphs, it is not universal for isometric embeddings of graphs, where an isometric embedding is a graph isomorphism which preserves distance. The Rado graph has diameter two, and so any graph with larger diameter does not embed isometrically into it. has described a family of universal graphs for isometric embedding, one for each possible finite graph diameter; the graph in his family with diameter two is the Rado graph. The Henson graphs are countable graphs (one for each positive integer ) that do not contain an -vertex clique, and are universal for -clique-free graphs.
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A 1-planar drawing of the Heawood graph: six of the edges have a single crossing, and the remaining 15 edges are not crossed. In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing point, where it crosses a single additional edge. If a 1-planar graph, one of the most natural generalizations of planar graphs, is drawn that way, the drawing is called a 1-plane graph or 1-planar embedding of the graph.
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The Biggs-Smith graph, the largest 3-regular distance-transitive graph. In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance, there is an automorphism of the graph that carries v to x and w to y. Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith. A distance-transitive graph is interesting partly because it has a large automorphism group.
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A grid graph is a unit distance graph corresponding to the square lattice, so that it is isomorphic to the graph having a vertex corresponding to every pair of integers (a, b), and an edge connecting (a, b) to (a+1, b) and (a, b+1). The finite grid graph Gm,n is an m×n rectangular graph isomorphic to the one obtained by restricting the ordered pairs to the range 0 ≤ a < m, 0 ≤ b < n. Grid graphs can be obtained as the Cartesian product of two paths: Gm,n = Pm × Pn. Every grid graph is a median graph.
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The square of a graph In graph theory, a branch of mathematics, the kth power Gk of an undirected graph G is another graph that has the same set of vertices, but in which two vertices are adjacent when their distance in G is at most k. Powers of graphs are referred to using terminology similar to that of exponentiation of numbers: G2 is called the square of G, G3 is called the cube of G, etc.. Graph powers should be distinguished from the products of a graph with itself, which (unlike powers) generally have many more vertices than the original graph.
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For instance, the intersection graph of line segments in one dimension is an interval graph; the intersection graph of unit disks in the plane is a unit disk graph. The Circle packing theorem states that the intersection graphs of non-crossing circles are exactly the planar graphs. Scheinerman's conjecture (proven in 2009) states that every planar graph can be represented as the intersection graph of line segments in the plane. A Levi graph of a family of points and lines has a vertex for each of these objects and an edge for every incident point-line pair.
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If such a graph exists, it would necessarily be a locally linear graph and a strongly regular graph with parameters (99,14,1,2). The first, third, and fourth parameters encode the statement of the problem: the graph should have 99 vertices, every pair of adjacent vertices should have 1 common neighbor, and every pair of non-adjacent vertices should have 2 common neighbors. The second parameter means that the graph is a regular graph with 14 edges per vertex. If this graph exists, it does not have any symmetries of order 11, which implies that its symmetries cannot take every vertex to every other vertex.
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In mathematics, a k-ultrahomogeneous graph is a graph in which every isomorphism between two of its induced subgraphs of at most k vertices can be extended to an automorphism of the whole graph. A k-homogeneous graph obeys a weakened version of the same property in which every isomorphism between two induced subgraphs implies the existence of an automorphism of the whole graph that maps one subgraph to the other (but does not necessarily extend the given isomorphism). A homogeneous graph is a graph that is k-homogeneous for every k, or equivalently k-ultrahomogeneous for every k.
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The Paley graph of order 9, colored with three colors and showing a clique of three vertices. In this graph and each of its induced subgraphs the chromatic number equals the clique number, so it is a perfect graph. In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph (clique number). Equivalently stated in symbolic terms an arbitrary graph G=(V,E) is perfect if and only if for all S\subseteq V we have \chi(G[S])=\omega(G[S]).
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A graph with eight vertices, and a tree decomposition of it onto a tree with six nodes. Each graph edge connects two vertices that are listed together at some tree node, and each graph vertex is listed at the nodes of a contiguous subtree of the tree. Each tree node lists at most three vertices, so the width of this decomposition is two. In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph.
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A self- complementary graph is a graph in which replacing every edge by a non-edge and vice versa produces an isomorphic graph. For instance, a five-vertex cycle graph is self-complementary, and is also a circulant graph. More generally every Paley graph of prime order is a self-complementary circulant graph.. Horst Sachs showed that, if a number has the property that every prime factor of is congruent to , then there exists a self-complementary circulant with vertices. He conjectured that this condition is also necessary: that no other values of allow a self-complementary circulant to exist.
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For instance, the diamond graph K1,1,2 (two triangles sharing an edge) has four graph automorphisms but its line graph K1,2,2 has eight. In the illustration of the diamond graph shown, rotating the graph by 90 degrees is not a symmetry of the graph, but is a symmetry of its line graph. However, all such exceptional cases have at most four vertices. A strengthened version of the Whitney isomorphism theorem states that, for connected graphs with more than four vertices, there is a one-to-one correspondence between isomorphisms of the graphs and isomorphisms of their line graphs.
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Every forest and every cactus graph are outerplanar., p. 169. The weak planar dual graph of an embedded outerplanar graph (the graph that has a vertex for every bounded face of the embedding, and an edge for every pair of adjacent bounded faces) is a forest, and the weak planar dual of a Halin graph is an outerplanar graph. A planar graph is outerplanar if and only if its weak dual is a forest, and it is Halin if and only if its weak dual is biconnected and outerplanar.. There is a notion of degree of outerplanarity.
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Distinguishing coloring of a 4-hypercube graph In graph theory, a distinguishing coloring or distinguishing labeling of a graph is an assignment of colors or labels to the vertices of the graph that destroys all of the nontrivial symmetries of the graph. The coloring does not need to be a proper coloring: adjacent vertices are allowed to be given the same color. For the colored graph, there should not exist any one-to-one mapping of the vertices to themselves that preserves both adjacency and coloring. The minimum number of colors in a distinguishing coloring is called the distinguishing number of the graph.
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It is also the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
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If the final graph consists of a single node, then the original graph is said to be reducible.
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The claw graph and the path graph on 4 vertices both have the same chromatic polynomial, for example.
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In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix.
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A climograph combines a line graph of mean monthly temperature with a bar graph of total monthly precipitation.
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A simple cycle separator in the dual graph of a planar graph forms an edge separator in the original graph.; . Applying the simple cycle separator theorem of to the dual graph of a given planar graph strengthens the O(√(Δn)) bound for the size of an edge separator by showing that every planar graph has an edge separator whose size is proportional to the Euclidean norm of its vector of vertex degrees. describe a polynomial time algorithm for finding the smallest edge separator that partitions a graph G into two subgraphs of equal size, when G is an induced subgraph of a grid graph with no holes or with a constant number of holes.
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A maximal outerplanar graph is a graph formed by a simple polygon in the plane by triangulating its interior. Every maximal outerplanar graph is pancyclic, as can be shown by induction. The outer face of the graph is an n-vertex cycle, and removing any triangle connected to the rest of the graph by only one edge (a leaf of the tree that forms the dual graph of the triangulation) forms a maximal outerplanar graph on one fewer vertex, that by induction has cycles of all the remaining lengths. With more care in choosing which triangle to remove, the same argument shows more strongly that every maximal outerplanar graph is node-pancyclic.
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Two red graphs are duals for the blue one, but they are not isomorphic. Because the dual graph depends on a particular embedding, the dual graph of a planar graph is not unique, in the sense that the same planar graph can have non-isomorphic dual graphs.. In the picture, the blue graphs are isomorphic but their dual red graphs are not. The upper red dual has a vertex with degree 6 (corresponding to the outer face of the blue graph) while in the lower red graph all degrees are less than 6. Hassler Whitney showed that if the graph is 3-connected then the embedding, and thus the dual graph, is unique.
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In graph theory, a branch of mathematics, the Moser spindle (also called the Mosers' spindle or Moser graph) is an undirected graph, named after mathematicians Leo Moser and his brother William,. with seven vertices and eleven edges. It is a unit distance graph requiring four colors in any graph coloring, and its existence can be used to prove that the chromatic number of the plane is at least four.. The Moser spindle has also been called the Hajós graph after György Hajós, as it can be viewed as an instance of the Hajós construction.. However, the name "Hajós graph" has also been applied to a different graph, in the form of a triangle inscribed within a hexagon..
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The Goldner–Harary graph, a planar graph with book thickness three The book thickness of a given graph is at most one if and only if is an outerplanar graph. An outerplanar graph is a graph that has a planar embedding in which all vertices belong to the outer face of the embedding. For such a graph, placing the vertices in the same order along the spine as they appear in the outer face provides a one-page book embedding of the given graph. (An articulation point of the graph will necessarily appear more than once in the cyclic ordering of vertices around the outer face, but only one of those copies should be included in the book embedding.) Conversely, a one-page book embedding is automatically an outerplanar embedding.
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The Möbius–Kantor graph is also a Cayley graph. The generalized Petersen graph G(n,k) is vertex-transitive if and only if n = 10 and k =2 or if k2 ≡ ±1 (mod n) and is edge-transitive only in the following seven cases: (n,k) = (4,1), (5,2), (8,3), (10,2), (10,3), (12,5), or (24,5) . So the Möbius–Kantor graph is one of only seven symmetric Generalized Petersen graphs. Its symmetric double torus embedding is correspondingly one of only seven regular cubic maps in which the total number of vertices is twice the number of vertices per face . Among the seven symmetric generalized Petersen graphs are the cubical graph G(4,1), the Petersen graph G(5,2), the dodecahedral graph G(10,2), the Desargues graph G(10,3) and the Nauru graph G(12,5).
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However, fewer colors may be obtained by forming an auxiliary graph that has a vertex for each vertex or face of the given planar graph, and in which two auxiliary graph vertices are adjacent whenever they correspond to adjacent features of the given planar graph. A vertex coloring of the auxiliary graph corresponds to a vertex-face coloring of the original planar graph. This auxiliary graph is 1-planar, from which it follows that Ringel's vertex-face coloring problem may also be solved with six colors. The graph K6 cannot be formed as an auxiliary graph in this way, but nevertheless the vertex-face coloring problem also sometimes requires six colors; for instance, if the planar graph to be colored is a triangular prism, then its eleven vertices and faces require six colors, because no three of them may be given a single color..
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Representing signals in frequency domain is a common approach to data compression. As graph signals can be sparse in their graph spectral domain, the graph Fourier transform can also be used for image compression.
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If a graph has diameter d, then its d-th power is the complete graph. If a graph family has bounded clique-width, then so do its d-th powers for any fixed d..
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Gary Theodore Chartrand (born 1936) is an American-born mathematician who specializes in graph theory. He is known for his textbooks on introductory graph theory and for the concept of a highly irregular graph.
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A polytree. In mathematics, and more specifically in graph theory, a polytree. (also called directed tree, oriented tree. or singly connected network.) is a directed acyclic graph whose underlying undirected graph is a tree.
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In graph theory, a graph or digraph whose adjacency matrix is circulant is called a circulant graph (or digraph). Equivalently, a graph is circulant if its automorphism group contains a full-length cycle. The Möbius ladders are examples of circulant graphs, as are the Paley graphs for fields of prime order.
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This graph is a 3-regular graph with 56 vertices and 84 edges, named after Felix Klein. It is a Hamiltonian graph. It has chromatic number 3, chromatic index 3, radius 6, diameter 6 and girth 7. It is also a 3-vertex-connected and a 3-edge-connected graph.
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There are three chapters on the four color theorem and graph coloring, a chapter on algebraic graph theory, and a final chapter on graph factorization. Appendices provide a brief update on graph history since 1936, biographies of the authors of the works included in the book, and a comprehensive bibliography.
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The strong perfect graph theorem of states that a graph is perfect if and only if none of its induced subgraphs are cycles of odd length greater than or equal to five, or their complements. Because this characterization is unaffected by graph complementation, it immediately implies the weak perfect graph theorem.
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If a graph is -tough, then one consequence (obtained by setting ) is that any set of nodes can be removed without splitting the graph in two. That is, every -tough graph is also -vertex-connected.
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The graph application is for manipulating directed and undirected graphs. Some the standard graph functions exist (like for adjacency and cliques) together with combinatorial functions like computing the lattice represented by a directed acyclic graph.
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The Robertson graph is the unique (4,5)-cage graph and was discovered by Robertson in 1964.Robertson, N. "The Smallest Graph of Girth 5 and Valency 4." Bull. Amer. Math. Soc. 70, 824-825, 1964.
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The perfect graph theorem states: :The complement of a perfect graph is perfect. Equivalently, in a perfect graph, the size of the maximum independent set equals the minimum number of cliques in a clique cover.
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With vertex 0, this graph is disconnected. The rest of the graph is connected. In an undirected graph , two vertices and are called connected if contains a path from to . Otherwise, they are called disconnected.
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In some reactive languages, the graph of dependencies is static, i.e., the graph is fixed throughout the program's execution. In other languages, the graph can be dynamic, i.e., it can change as the program executes.
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The stochastic version of kronecker graph eliminates the staircase effect, which happens due to large multiplicity of kronecker graph.
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The dual of such a graph cannot exist, but is the graph required to represent a generalised mesh elimination.
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Outermost graph reduction is referred to as lazy evaluation and innermost graph reduction is referred to as eager evaluation.
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13, 231-237, 1970. The smallest half-transitive graph is the Holt graph, with degree 4 and 27 vertices..
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Unlike later graphs in this sequence, the Grötzsch graph is the smallest triangle-free graph with its chromatic number .
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Defining an ‘irregular graph’ was not immediately obvious. In a k-regular graph, all vertices have degree k. In any graph G with more than one vertex, two vertices in G must have the same degree, so an irregular graph cannot be defined as a graph with all vertices of different degrees. One may be tempted then to define an irregular graph as having all vertices of distinct degrees except for two, but these types of graphs are also well understood and thus not interesting.
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A nested triangles graph with 18 vertices In graph theory, a nested triangles graph with n vertices is a planar graph formed from a sequence of n/3 triangles, by connecting pairs of corresponding vertices on consecutive triangles in the sequence. It can also be formed geometrically, by gluing together n/3 − 1 triangular prisms on their triangular faces. This graph, and graphs closely related to it, have been frequently used in graph drawing to prove lower bounds on the area requirements of various styles of drawings.
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Extremal graph theory is a branch of mathematics that studies how global properties of a graph influence local substructure. It encompasses a vast number of results that describe how do certain graph properties - number of vertices (size), number of edges, edge density, chromatic number, and girth, for example - guarantee the existence of certain local substructures.The Turán graph T(n,r) is an example of an extremal graph. It has the maximum possible number of edges for a graph on n vertices without (r + 1)-cliques.
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The Folkman graph, the smallest semi-symmetric graph. In the mathematical field of graph theory, a semi-symmetric graph is an undirected graph that is edge-transitive and regular, but not vertex-transitive. In other words, a graph is semi-symmetric if each vertex has the same number of incident edges, and there is a symmetry taking any of the graph's edges to any other of its edges, but there is some pair of vertices such that no symmetry maps the first into the second.
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The Coxeter graph is also uniquely determined by its graph spectrum, the set of graph eigenvalues of its adjacency matrix.E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189-202, 2003 As a finite connected vertex- transitive graph that contains no Hamiltonian cycle, the Coxeter graph is a counterexample to a variant of the Lovász conjecture, but the canonical formulation of the conjecture asks for an Hamiltonian path and is verified by the Coxeter graph.
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In the mathematical field of graph theory, the Gray graph is an undirected bipartite graph with 54 vertices and 81 edges. It is a cubic graph: every vertex touches exactly three edges. It was discovered by Marion C. Gray in 1932 (unpublished), then discovered independently by Bouwer 1968 in reply to a question posed by Jon Folkman 1967. The Gray graph is interesting as the first known example of a cubic graph having the algebraic property of being edge but not vertex transitive (see below).
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In graph theory and computer science, the graph sandwich problem is a problem of finding a graph that belongs to a particular family of graphs and is "sandwiched" between two other graphs, one of which must be a subgraph and the other of which must be a supergraph of the desired graph. Graph sandwich problems generalize the problem of testing whether a given graph belongs to a family of graphs, and have attracted attention because of their applications and as a natural generalization of recognition problems..
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A graph is a circle graph if and only if it is the overlap graph of a set of intervals on a line. This is a graph in which the vertices correspond to the intervals, and two vertices are connected by an edge if the two intervals overlap, with neither containing the other. The intersection graph of a set of intervals on a line is called the interval graph. String graphs, the intersection graphs of curves in the plane, include circle graphs as a special case.
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In graph theory, Turán's theorem is a result on the number of edges in a Kr+1-free graph. An -vertex graph that does not contain any -vertex clique may be formed by partitioning the set of vertices into parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong to two different parts. The resulting graph is the Turán graph . Turán's theorem states that the Turán graph has the largest number of edges among all -free -vertex graphs.
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In a biconnected graph of circuit rank less than three (such as a cycle graph or theta graph) every cycle is peripheral, but every biconnected graph with circuit rank three or more has a non-peripheral cycle, which may be found in linear time., Lemma 3.4, pp. 75–76. Generalizing chordal graphs, define a strangulated graph to be a graph in which every peripheral cycle is a triangle. They characterize these graphs as being the clique-sums of chordal graphs and maximal planar graphs..
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Here, we use properties of graph entropy to provide a simple proof that a complete graph G on n vertices cannot be expressed as the union of fewer than \log_2 n bipartite graphs. Proof By monotonicity, no bipartite graph can have graph entropy greater than that of a complete bipartite graph, which is bounded by 1. Thus, by sub-additivity, the union of k bipartite graphs cannot have entropy greater than k. Now let G = (V, E) be a complete graph on n vertices.
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A graph and two of its cuts. The dotted line in red represents a cut with three crossing edges. The dashed line in green represents one of the minimum cuts of this graph, crossing only two edges. In graph theory, a minimum cut or min-cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets) that is minimal in some sense.
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Equivalently, every -critical graph (a graph that requires colors but for which every proper subgraph requires fewer colors) is -constructible.A proof can also be found in . Alternatively, every graph that requires colors may be formed by combining the Hajós construction, the operation of identifying any two nonadjacent vertices, and the operations of adding a vertex or edge to the given graph, starting from the complete graph ., p. 184.
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Several improvements to graph databases appeared in the early 1990s, accelerating in the late 1990s with endeavors to index web pages. In the mid-to-late 2000s, commercial graph databases with ACID guarantees such as Neo4j and Oracle Spatial and Graph became available. In the 2010s, commercial ACID graph databases that could be scaled horizontally became available. Further, SAP HANA brought in-memory and columnar technologies to graph databases.
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The Goldner–Harary graph is a planar graph: it can be drawn in the plane with none of its edges crossing. When drawn on a plane, all its faces are triangular, making it a maximal planar graph. As with every maximal planar graph, it is also 3-vertex-connected: the removal of any two of its vertices leaves a connected subgraph. The Goldner–Harary graph is also non-hamiltonian.
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In the mathematical field of graph theory, the Harries graph or Harries (3-10)-cage is a 3-regular undirected graph with 70 vertices and 105 edges. The Harries graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 and is Hamiltonian. It is also a 3-vertex-connected and 3-edge-connected non-planar cubic graph. It has book thickness 3 and queue number 2.
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Rado made contributions in combinatorics and graph theory including 18 papers with Paul Erdős. In graph theory, the Rado graph, a countably infinite graph containing all countably infinite graphs as induced subgraphs, is named after Rado. He rediscovered it in 1964 after previous works on the same graph by Wilhelm Ackermann, Paul Erdős, and Alfréd Rényi. In combinatorial set theory, the Erdős–Rado theorem extends Ramsey's theorem to infinite sets.
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In graph-theoretic mathematics, a biregular graph. or semiregular bipartite graph. is a bipartite graph G=(U,V,E) for which every two vertices on the same side of the given bipartition have the same degree as each other. If the degree of the vertices in U is x and the degree of the vertices in V is y, then the graph is said to be (x,y)-biregular.
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The Young–Fibonacci lattice is an infinite modular lattice having these digit sequences as its elements, compatible with this rank structure. The Young–Fibonacci graph is the graph of this lattice, and has a vertex for each digit sequence. As the graph of a modular lattice, it is a modular graph. The Young–Fibonacci graph and the Young–Fibonacci lattice were both initially studied in two papers by and .
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Walecki's Hamiltonian decomposition of the complete graph K_9 In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles. Hamiltonian decompositions have been studied both for undirected graphs and for directed graphs; in the undirected case, a Hamiltonian decomposition can also be described as a 2-factorization of the graph such that each factor is connected.
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4\times 4 Sudoku graph In the mathematics of Sudoku, the Sudoku graph is an undirected graph whose vertices represent the cells of a (blank) Sudoku puzzle and whose edges represent pairs of cells that belong to the same row, column, or block of the puzzle. The problem of solving a Sudoku puzzle can be represented as precoloring extension on this graph. It is an integral Cayley graph.
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In graph theory, a perfectly orderable graph is a graph whose vertices can be ordered in such a way that a greedy coloring algorithm with that ordering optimally colors every induced subgraph of the given graph. Perfectly orderable graphs form a special case of the perfect graphs, and they include the chordal graphs, comparability graphs, and distance-hereditary graphs. However, testing whether a graph is perfectly orderable is NP-complete.
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The Sudoku graph contains as a subgraph the rook's graph, which is defined in the same way using only the rows and columns (but not the blocks) of the Sudoku board. The 20-regular 81-vertex Sudoku graph should be distinguished from a different 20-regular graph on 81 vertices, the Brouwer–Haemers graph, which has smaller cliques (of size 3) and requires fewer colors (7 instead of 9).
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In the mathematical field of graph theory, the Meredith graph is a 4-regular undirected graph with 70 vertices and 140 edges discovered by Guy H. J. Meredith in 1973. The Meredith graph is 4-vertex-connected and 4-edge- connected, has chromatic number 3, chromatic index 5, radius 7, diameter 8, girth 4 and is non-hamiltonian.Bondy, J. A. and Murty, U. S. R. "Graph Theory". Springer, p.
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In the mathematical field of graph theory, the Horton graph or Horton 96-graph is a 3-regular graph with 96 vertices and 144 edges discovered by Joseph Horton. Published by Bondy and Murty in 1976, it provides a counterexample to the Tutte conjecture that every cubic 3-connected bipartite graph is Hamiltonian.Tutte, W. T. "On the 2-Factors of Bicubic Graphs." Discrete Math. 1, 203-208, 1971/72.
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The theorem has particular use in algebraic graph theory. The "underlying graph" of a nonnegative n-square matrix is the graph with vertices numbered 1, ..., n and arc ij if and only if Aij ≠ 0. If the underlying graph of such a matrix is strongly connected, then the matrix is irreducible, and thus the theorem applies. In particular, the adjacency matrix of a strongly connected graph is irreducible.
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In the mathematical field of graph theory, the Folkman graph, named after Jon Folkman, is a bipartite 4-regular graph with 20 vertices and 40 edges. The Folkman graph is Hamiltonian and has chromatic number 2, chromatic index 4, radius 3, diameter 4 and girth 4\. It is also a 4-vertex-connected and 4-edge- connected perfect graph. It has book thickness 3 and queue number 2.
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Gain graphs used in topological graph theory as a means to construct graph embeddings in surfaces are known as "voltage graphs" (Gross 1974; Gross and Tucker 1977). The term "gain graph" is more usual in other contexts, e.g., biased graph theory and matroid theory. The term group-labelled graph has also been used, but it is ambiguous since "group labels" may be--and have been--treated as weights.
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In a maximal planar graph, or more generally in every polyhedral graph, the peripheral cycles are exactly the faces of a planar embedding of the graph, so a polyhedral graph is strangulated if and only if all the faces are triangles, or equivalently it is maximal planar. Every chordal graph is strangulated, because the only induced cycles in chordal graphs are triangles, so there are no longer cycles to delete.
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Other results and conjectures involving graph minors include the graph structure theorem, according to which the graphs that do not have H as a minor may be formed by gluing together simpler pieces, and Hadwiger's conjecture relating the inability to color a graph to the existence of a large complete graph as a minor of it. Important variants of graph minors include the topological minors and immersion minors.
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In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces.J.L. Gross and T.W. Tucker, Topological graph theory, Wiley Interscience, 1987 It also studies immersions of graphs. Embedding a graph in a surface means that we want to draw the graph on a surface, a sphere for example, without two edges intersecting.
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A Hamiltonian path or traceable path is a path that visits each vertex of the graph exactly once. A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once.
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In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, n-1.
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Every quotient graph has one of three forms: it may be a prime graph, a complete graph, or a star. A graph may have exponentially many different splits, but they are all represented in the split decomposition tree, either as an edge of the tree (for a strong split) or as an arbitrary partition of a complete or star quotient graph (for a split that is not strong).
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If such a walk exists, the graph is called traversable or semi-eulerian.Jun- ichi Yamaguchi, Introduction of Graph Theory. An Eulerian cycle, Eulerian circuit or Euler tour in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal.
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Coin graphs are always connected, simple, and planar. The circle packing theorem states that these are the only requirements for a graph to be a coin graph: Circle packing theorem: For every connected simple planar graph G there is a circle packing in the plane whose intersection graph is (isomorphic to) G.
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Another of these five graphs is the generalized Petersen graph , also known as the Nauru graph. . The McGee graph has radius 4, diameter 4, chromatic number 3 and chromatic index 3. It is also a 3-vertex-connected and a 3-edge-connected graph. It has book thickness 3 and queue number 2.
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Apex-minor- free graph families obey a strengthened version of the graph structure theorem, leading to additional approximation algorithms for graph coloring and the travelling salesman problem.. However, some of these results can also be extended to arbitrary minor-closed graph families via structure theorems relating them to apex-minor-free graphs..
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An undirected graph that is not connected is called disconnected. An undirected graph G is therefore disconnected if there exist two vertices in G such that no path in G has these vertices as endpoints. A graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected.
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It is possible in linear time to test whether a graph is a Hamming graph, and in the case that it is, find a labeling of it with tuples that realizes it as a Hamming graph.
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Every regular bipartite graph is also biregular. Every edge-transitive graph (disallowing graphs with isolated vertices) that is not also vertex-transitive must be biregular.. In particular every edge-transitive graph is either regular or biregular.
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A skew-symmetric graph may equivalently be defined as the double covering graph of a polar graph (introduced by , , called a switch graph by ), which is an undirected graph in which the edges incident to each vertex are partitioned into two subsets. Each vertex of the polar graph corresponds to two vertices of the skew-symmetric graph, and each edge of the polar graph corresponds to two edges of the skew-symmetric graph. This equivalence is the one used by to model problems of matching in terms of skew-symmetric graphs; in that application, the two subsets of edges at each vertex are the unmatched edges and the matched edges. Zelinka (following F. Zitek) and Cook visualize the vertices of a polar graph as points where multiple tracks of a train track come together: if a train enters a switch via a track that comes in from one direction, it must exit via a track in the other direction.
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The Rado graph, as numbered by and . In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge. The names of this graph honor Richard Rado, Paul Erdős, and Alfréd Rényi, mathematicians who studied it in the early 1960s; it appears even earlier in the work of . The Rado graph can also be constructed non- randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of the natural numbers, or as an infinite Paley graph that has edges connecting pairs of prime numbers congruent to 1 mod 4 that are quadratic residues modulo each other.
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Tait conjectured that every cubic polyhedral graph (that is, a polyhedral graph in which each vertex is incident to exactly three edges) has a Hamiltonian cycle, but this conjecture was disproved by a counterexample of W. T. Tutte, the polyhedral but non-Hamiltonian Tutte graph. If one relaxes the requirement that the graph be cubic, there are much smaller non- Hamiltonian polyhedral graphs. The graph with the fewest vertices and edges is the 11-vertex and 18-edge Herschel graph,. and there also exists an 11-vertex non-Hamiltonian polyhedral graph in which all faces are triangles, the Goldner–Harary graph.. More strongly, there exists a constant α < 1 (the shortness exponent) and an infinite family of polyhedral graphs such that the length of the longest simple path of an n-vertex graph in the family is O(nα)...
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It is hypohamiltonian, meaning that although it has no Hamiltonian cycle, deleting any vertex makes it Hamiltonian, and is the smallest hypohamiltonian graph. As a finite connected vertex-transitive graph that does not have a Hamiltonian cycle, the Petersen graph is a counterexample to a variant of the Lovász conjecture, but the canonical formulation of the conjecture asks for a Hamiltonian path and is verified by the Petersen graph. Only five connected vertex-transitive graphs with no Hamiltonian cycles are known: the complete graph K2, the Petersen graph, the Coxeter graph and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle.Royle, G. "Cubic Symmetric Graphs (The Foster Census)." If G is a 2-connected, r-regular graph with at most 3r + 1 vertices, then G is Hamiltonian or G is the Petersen graph.
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The graphs of branchwidth 2 are the graphs in which each biconnected component is a series-parallel graph; the only minimal forbidden minor is the complete graph K4 on four vertices., Theorem 4.2, p. 165. A graph has branchwidth three if and only if it has treewidth three and does not have the cube graph as a minor; therefore, the four minimal forbidden minors are three of the four forbidden minors for treewidth three (the graph of the octahedron, the complete graph K5, and the Wagner graph) together with the cube graph.. The fourth forbidden minor for treewidth three, the pentagonal prism, has the cube graph as a minor, so it is not minimal for branchwidth three. Forbidden minors have also been studied for matroid branchwidth, despite the lack of a full analogue to the Robertson–Seymour theorem in this case.
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The cut induced by such a partition in the dual graph corresponds to a Hamiltonian cycle in the primal graph.
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Guy (2004) pp.190–191 The seven-page book graph provides an example of a graph that is not harmonious..
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Master Thesis, University of Tübingen, 2018 The Robertson graph is also a Hamiltonian graph which possesses distinct directed Hamiltonian cycles.
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In graph theory, the treewidth of an undirected graph is a number associated with the graph. Treewidth may be defined in several equivalent ways: the size of the largest vertex set in a tree decomposition of the graph, the size of the largest clique in a chordal completion of the graph, the maximum order of a haven describing a strategy for a pursuit-evasion game on the graph, or the maximum order of a bramble, a collection of connected subgraphs that all touch each other. Treewidth is commonly used as a parameter in the parameterized complexity analysis of graph algorithms. The graphs with treewidth at most k are also called partial k-trees; many other well-studied graph families also have bounded treewidth.
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To construct these graphs, Henson orders the vertices of the Rado graph into a sequence with the property that, for every finite set of vertices, there are infinitely many vertices having as their set of earlier neighbors. (The existence of such a sequence uniquely defines the Rado graph.) He then defines to be the induced subgraph of the Rado graph formed by removing the final vertex (in the sequence ordering) of every -clique of the Rado graph. With this construction, each graph is an induced subgraph of , and the union of this chain of induced subgraphs is the Rado graph itself. Because each graph omits at least one vertex from each -clique of the Rado graph, there can be no -clique in .
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Constructing a penny graph from the locations of its circles can be performed as an instance of the closest pair of points problem, taking worst-case time or (with randomized time and with the use of the floor function) expected time .. An alternative method with the same worst-case time is to construct the Delaunay triangulation or nearest neighbor graph of the circle centers (both of which contain the penny graph as a subgraph) and then test which edges correspond to circle tangencies. However, testing whether a given graph is a penny graph is NP-hard, even when the given graph is a tree.. Similarly, testing whether a graph is a three-dimensional mutual nearest neighbor graph is also NP-hard.
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In planar graphs, the set of bounded cycles of an embedding of the graph forms a cycle basis. The minimum weight cycle basis of a planar graph corresponds to the Gomory–Hu tree of the dual graph.
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Every chordal bipartite graph is a modular graph. The chordal bipartite graphs include the complete bipartite graphs and the bipartite distance-hereditary graphs.Chordal bipartite graphs, Information System on Graph Classes and their Inclusions, retrieved 2016-09-30.
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Marion Gray (26 March 1902 – 16 September 1979) was a Scottish mathematician who discovered a graph with 54 vertices and 81 edges while working at American Telephone & Telegraph. The graph is commonly known as the Gray graph.
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It is isomorphic to the 16-vertex hypercube graph Q4. A closely related configuration, the Möbius–Kantor configuration formed by two mutually inscribed quadrilaterals, has the Möbius–Kantor graph, a subgraph of Q4, as its Levi graph.
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If a graph G is formed from the Rado graph by deleting any finite number of edges or vertices, or adding a finite number of edges, the change does not affect the extension property of the graph. For any pair of sets U and V it is still possible to find a vertex in the modified graph that is adjacent to everything in U and nonadjacent to everything in V, by adding the modified parts of G to V and applying the extension property in the unmodified Rado graph. Therefore, any finite modification of this type results in a graph that is isomorphic to the Rado graph., Section 1.3: Indestructibility.
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In the mathematical field of graph theory, a zero-symmetric graph is a connected graph in which each vertex has exactly three incident edges and, for each two vertices, there is a unique symmetry taking one vertex to the other. Such a graph is a vertex-transitive graph but cannot be an edge-transitive graph: the number of symmetries equals the number of vertices, too few to take every edge to every other edge. The smallest zero-symmetric graph with two edge orbits The name for this class of graphs was coined by R. M. Foster in a 1966 letter to H. S. M. Coxeter., p. ix.
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The Petersen graph is a unit distance graph: each inner vertex is rotated 90^\circ from its adjacent outer vertex relative to the center of the drawing. In mathematics, and particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points by an edge whenever the distance between the two points is exactly one. Edges of unit distance graphs sometimes cross each other, so they are not always planar; a unit distance graph without crossings is called a matchstick graph. The Hadwiger–Nelson problem concerns the chromatic number of unit distance graphs.
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By Steinitz's theorem, these graphs are exactly the polyhedral graphs, the graphs of convex polyhedra. A planar graph is 3-vertex-connected if and only if its dual graph is 3-vertex-connected. More generally, a planar graph has a unique embedding, and therefore also a unique dual, if and only if it is a subdivision of a 3-vertex-connected planar graph (a graph formed from a 3-vertex-connected planar graph by replacing some of its edges by paths). For some planar graphs that are not 3-vertex- connected, such as the complete bipartite graph , the embedding is not unique, but all embeddings are isomorphic.
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When the algorithm completes, data structure will actually describe a graph that is a subset of the original graph with some edges removed. Its key property will be that if the algorithm was run with some starting node, then every path from that node to any other node in the new graph will be the shortest path between those nodes in the original graph, and all paths of that length from the original graph will be present in the new graph. Then to actually find all these shortest paths between two given nodes we would use a path finding algorithm on the new graph, such as depth-first search.
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The Petersen graph, shown below, is a bivariegated graph: if one partitions it into an outer pentagon and an inner five-point star, each vertex on one side of the partition has exactly one neighbor on the other side of the partition. More generally, the same is true for any generalized Petersen graph formed by connecting an outer polygon and an inner star with the same number of points; for instance, this applies to the Möbius–Kantor graph and the Desargues graph. 150px Any hypercube graph, such as the four-dimensional hypercube shown below, is also bivariegated. 150px However, the graph shown below is not bivariegated.
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The generalized Petersen graphs also include the n-prism G(n,1) the Dürer graph G(6,2), the Möbius-Kantor graph G(8,3), the dodecahedron G(10,2), the Desargues graph G(10,3) and the Nauru graph G(12,5). The Petersen family consists of the seven graphs that can be formed from the Petersen graph by zero or more applications of Δ-Y or Y-Δ transforms. The complete graph K6 is also in the Petersen family. These graphs form the forbidden minors for linklessly embeddable graphs, graphs that can be embedded into three-dimensional space in such a way that no two cycles in the graph are linked.
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Let be a family of sets (allowing sets in to be repeated); then the intersection graph of is an undirected graph that has a vertex for each member of and an edge between each two members that have a nonempty intersection. Every graph can be represented as an intersection graph in this way.. The intersection number of the graph is the smallest number such that there exists a representation of this type for which the union of has elements. The problem of finding an intersection representation of a graph with a given number of elements is known as the intersection graph basis problem., Problem GT59.
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A simple directed graph. Here the double-headed arrow represents two distinct edges, one for each direction. In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by edges, where the edges have a direction associated with them.
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The composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph. In the opposite direction, by Frucht's theorem, all groups can be represented as the automorphism group of a connected graph – indeed, of a cubic graph...
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Both of the two types of stabilizers are maximal subgroups of the whole automorphism group of the Hoffman–Singleton graph. The characteristic polynomial of the Hoffman–Singleton graph is equal to (x-7) (x-2)^{28} (x+3)^{21}. Therefore, the Hoffman–Singleton graph is an integral graph: its spectrum consists entirely of integers.
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There are 100 independent sets of size 15 in the Hoffman–Singleton graph. Create a new graph with 100 corresponding vertices, and connect vertices whose corresponding independent sets have exactly 0 or 8 elements in common. The resulting Higman–Sims graph can be partitioned into two copies of the Hoffman-Singleton graph in 352 ways.
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Frank Harary and Dorwin Cartwright looked at Heider's triads as 3-cycles in a signed graph. The sign of a path in a graph is the product of the signs of its edges. They considered cycles in a signed graph representing a social network. : A balanced signed graph has only cycles of positive signs.
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The 6-vertex crown graph forms a cycle, and the 8-vertex crown graph is isomorphic to the graph of a cube. In the Schläfli double six, a configuration of 12 lines and 30 points in three-dimensional space, the twelve lines intersect each other in the pattern of a 12-vertex crown graph.
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A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.
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In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2. In other words, a graph is edge-transitive if its automorphism group acts transitively upon its edges.
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The barycentric subdivision subdivides each edge of the graph. This is a special subdivision, as it always results in a bipartite graph. This procedure can be repeated, so that the nth barycentric subdivision is the barycentric subdivision of the n−1th barycentric subdivision of the graph. The second such subdivision is always a simple graph.
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Construction of a trivially perfect graph from nested intervals and from the reachability relationship in a tree In graph theory, a trivially perfect graph is a graph with the property that in each of its induced subgraphs the size of the maximum independent set equals the number of maximal cliques., definition 2.6.2, p.34; .
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A planar graph is an undirected graph that can be embedded into the Euclidean plane without any crossings. A planar graph is called polyhedral if and only if it is 3-vertex-connected, that is, if there do not exist two vertices the removal of which would disconnect the rest of the graph. A graph is bipartite if its vertices can be colored with two different colors such that each edge has one endpoint of each color. A graph is cubic (or 3-regular) if each vertex is the endpoint of exactly three edges.
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W. T. Tutte conjectured that every snark has the Petersen graph as a minor. That is, he conjectured that the smallest snark, the Petersen graph, may be formed from any other snark by contracting some edges and deleting others. Equivalently (because the Petersen graph has maximum degree three) every snark has a subgraph that can be formed from the Petersen graph by subdividing some of its edges. This conjecture is a strengthened form of the four color theorem, because any graph containing the Petersen graph as a minor must be nonplanar.
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The graph automorphism problem is the problem of testing whether a graph has a nontrivial automorphism. It belongs to the class NP of computational complexity. Similar to the graph isomorphism problem, it is unknown whether it has a polynomial time algorithm or it is NP- complete.. There is a polynomial time algorithm for solving the graph automorphism problem for graphs where vertex degrees are bounded by a constant. The graph automorphism problem is polynomial-time many-one reducible to the graph isomorphism problem, but the converse reduction is unknown.
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Likewise, some authors use the term "flow graph" to refer strictly to the Coates flow graph. According to Henley & Williams: :"The nomenclature is far from standardized, and...no standardization can be expected in the foreseeable future." A designation "flow graph" that includes both the Mason graph and the Coates graph, and a variety of other forms of such graphs appears useful, and agrees with Abrahams and Coverley's and with Henley and Williams' approach. A directed network – also known as a flow network – is a particular type of flow graph.
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The Andrásfai graph And() for any natural number n \geq 1 is a circulant graph on 3n-1 vertices, in which vertex k is connected by an edge to vertices k\pm j, for every j that is congruent to 1 mod 3. For instance, the Wagner graph is an Andrásfai graph, the graph And(3). The graph family is triangle-free, and And() has an independence number of n. From this the formula R(3,n) \geq 3(n-1) results, where R(n,k) is the Ramsey number.
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A map with twelve pentagonal faces In topology and graph theory, a map is a subdivision of a surface such as the Euclidean plane into interior-disjoint regions, formed by embedding a graph onto the surface and forming connected components (faces) of the complement of the graph. That is, it is a tessellation of the surface. A map graph is a graph derived from a map by creating a vertex for each face and an edge for each pair of faces that meet at a vertex or edge of the embedded graph.
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The smallest connected half-transitive graph is Holt's graph, with degree 4 and 27 vertices.. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex-transitive and edge-transitive, rather than an arc-transitive graph. Such a definition would include half-transitive graphs, which are excluded under the definition above. A distance-transitive graph is one where instead of considering pairs of adjacent vertices (i.e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart.
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Since every apex graph is linkless embeddable, this shows that there are graphs that are linkless embeddable but not YΔY-reducible and therefore that there are additional forbidden minors for the YΔY-reducible graphs. Robertson's apex graph is shown in the figure. It can be obtained by connecting an apex vertex to each of the degree-three vertices of a rhombic dodecahedron, or by merging two diametrally opposed vertices of a four-dimensional hypercube graph. Because the rhombic dodecahedron's graph is planar, Robertson's graph is an apex graph.
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For instance, the Grötzsch graph, the Mycielskian of a five-vertex cycle- graph, is factor-critical.. Every 2-vertex-connected claw-free graph with an odd number of vertices is factor-critical. For instance, the 11-vertex graph formed by removing a vertex from the regular icosahedron (the graph of the gyroelongated pentagonal pyramid) is both 2-connected and claw-free, so it is factor-critical. This result follows directly from the more fundamental theorem that every connected claw-free graph with an even number of vertices has a perfect matching..
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In mathematics, Scheinerman's conjecture, now a theorem, states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis (1984), following earlier results that every planar graph could be represented as the intersection graph of a set of simple curves in the plane . It was proven by . For instance, the graph G shown below to the left may be represented as the intersection graph of the set of segments shown below to the right.
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In a line graph L(G), each vertex of degree k in the original graph G creates k(k − 1)/2 edges in the line graph. For many types of analysis this means high-degree nodes in G are over-represented in the line graph L(G). For instance, consider a random walk on the vertices of the original graph G. This will pass along some edge e with some frequency f. On the other hand, this edge e is mapped to a unique vertex, say v, in the line graph L(G).
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Schedule compliance with conflict serializability can be tested with the precedence graph (serializability graph, serialization graph, conflict graph) for committed transactions of the schedule. It is the directed graph representing precedence of transactions in the schedule, as reflected by precedence of conflicting operations in the transactions. :In the precedence graph transactions are nodes and precedence relations are directed edges. There exists an edge from a first transaction to a second transaction, if the second transaction is in conflict with the first (see Conflict serializability above), and the conflict is materialized (i.e.
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This directed graph has no cycles: it is not possible to get from any vertex (point) back to that same point, following the connections in the direction indicated by the arrows. A feedback arc set (FAS) or feedback edge set is a set of edges which, when removed from the graph, leaves an acyclic graph. Put another way, it is a set containing at least one edge of every cycle in the graph. In graph theory, a directed graph may contain directed cycles, a closed one-way path of edges.
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Frequently trees are assumed to have only one root. Note that trees in set theory are often defined to grow downward making the root the greatest node. Trees with a single root may be viewed as rooted trees in the sense of graph theory in one of two ways: either as a tree (graph theory) or as a trivially perfect graph. In the first case, the graph is the undirected Hasse Diagram of the partially ordered set, and in the second case, the graph is simply the underlying (undirected) graph of the partially ordered set.
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The 5-regular Clebsch graph is a strongly regular graph of degree 5 with parameters (v,k,\lambda,\mu) = (16, 5, 0, 2).Peter J. Cameron Strongly regular graphs on DesignTheory.org, 2001 Its complement, the 10-regular Clebsch graph, is therefore also a strongly regular graph, with parameters (16, 10, 6, 6). The 5-regular Clebsch graph is hamiltonian, non planar and non eulerian. It is also both 5-vertex-connected and 5-edge-connected.
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A directed graph is weakly connected (or just connected p. 19 in the 2007 edition; p. 20 in the 2nd edition (2009).) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph. A directed graph is strongly connected or strong if it contains a directed path from x to y and a directed path from y to x for every pair of vertices }.
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In mathematics and computer science, graph edit distance (GED) is a measure of similarity (or dissimilarity) between two graphs. The concept of graph edit distance was first formalized mathematically by Alberto Sanfeliu and King-Sun Fu in 1983. A major application of graph edit distance is in inexact graph matching, such as error-tolerant pattern recognition in machine learning. The graph edit distance between two graphs is related to the string edit distance between strings.
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A regular edge-transitive graph G cannot admit perfect state transfer between a pair of adjacent vertices, unless it is a disjoint union of copies of the complete graph K_2. A strongly regular graph admits perfect state transfer if and only if it is the complement of the disjoint union of an even number of copies of K_2. The only cubic distance-regular graph that admits perfect state transfer is the cubical graph.
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Every finite forest has either an isolated vertex (incident to no edges) or a leaf vertex (incident to exactly one edge); therefore, trees and forests are 1-degenerate graphs. Every 1-degenerate graph is a forest. Every finite planar graph has a vertex of degree five or less; therefore, every planar graph is 5-degenerate, and the degeneracy of any planar graph is at most five. Similarly, every outerplanar graph has degeneracy at most two,.
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As with any bipartite graph, there are no odd-length cycles, and there are also no cycles of four or six vertices, so the girth of the Gray graph is 8. The simplest oriented surface on which the Gray graph can be embedded has genus 7 . The Gray graph is Hamiltonian and can be constructed from the LCF notation: :[-25,7,-7,13,-13,25]^9.\ As a Hamiltonian cubic graph, it has chromatic index three.
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See . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees.
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In the mathematical field of graph theory, the Tutte graph is a 3-regular graph with 46 vertices and 69 edges named after W. T. Tutte. It has chromatic number 3, chromatic index 3, girth 4 and diameter 8. The Tutte graph is a cubic polyhedral graph, but is non-hamiltonian. Therefore, it is a counterexample to Tait's conjecture that every 3-regular polyhedron has a Hamiltonian cycle.. Reprinted in Scientific Papers, Vol.
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Every perfectly orderable graph is a perfect graph. Chordal graphs are perfectly orderable; a perfect ordering of a chordal graph may be found by reversing a perfect elimination ordering for the graph. Thus, applying greedy coloring to a perfect ordering provides an efficient algorithm for optimally coloring chordal graphs. Comparability graphs are also perfectly orderable, with a perfect ordering being given by a topological ordering of a transitive orientation of the graph.
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The Gray graph is edge-transitive and regular, but not vertex- transitive. Edge-transitive graphs include any complete bipartite graph K_{m,n}, and any symmetric graph, such as the vertices and edges of the cube. Symmetric graphs are also vertex-transitive (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. The Gray graph is an example of a graph which is edge-transitive but not vertex-transitive.
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The triangle finding problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph. It is possible to test whether a graph with m edges is triangle-free in time O(m1.41). Another approach is to find the trace of A3, where A is the adjacency matrix of the graph.
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In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic. A polytree is an example of an oriented graph.
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On the left a set of polygons inscribed in a circle; on the right the relative Polygon-circle graph (intersection graph of the polygons). On the bottom the alternating sequence of polygons around the circle. In the mathematical discipline of graph theory, a polygon-circle graph is an intersection graph of a set of convex polygons all of whose vertices lie on a common circle. These graphs have also been called spider graphs.
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Herbert Grötzsch (right) on his 86th birthday in Halle, with Camillo Herbert Grötzsch (21 May 1902 – 15 May 1993) was a German mathematician. He was born in Döbeln and died in Halle. Grötzsch worked in graph theory. He was the discoverer and eponym of the Grötzsch graph, a triangle-free graph that requires four colors in any graph coloring, and Grötzsch's theorem, the result that every triangle-free planar graph requires at most three colors.
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An abstract dependency graph is a directed graph, a graph of vertices connected by one-way edges. Most often, the vertices and edges of the graph represent the inputs and outputs of functions in or components of the system. By inspecting the created abstract dependency graph, the developer can detect syntactic anomalies (or Preece anomalies) in the system. While anomalies are not always defects, they often provide clues to finding defects in a system.
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Tutte (3,8)-cage. In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an (r,g)-graph is defined to be a graph in which each vertex has exactly r neighbors, and in which the shortest cycle has length exactly g. It is known that an (r,g)-graph exists for any combination of r ≥ 2 and g ≥ 3.
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In graph theory, a branch of mathematics, the Hajós construction is an operation on graphs named after that may be used to construct any critical graph or any graph whose chromatic number is at least some given threshold.
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J. Combin. 14, 397-407, 1993. The characteristic polynomial of the Higman–Sims graph is (x − 22)(x − 2)77(x + 8)22. Therefore the Higman–Sims graph is an integral graph: its spectrum consists entirely of integers.
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The odd graph O4 = KG7,3 In the mathematical field of graph theory, the odd graphs On are a family of symmetric graphs with high odd girth, defined from certain set systems. They include and generalize the Petersen graph.
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Transitive orientability of interval graph complements was proven by ; the characterization of interval graphs is due to . See also , prop. 1.3, pp. 15–16. A permutation graph is a containment graph on a set of intervals.. , theorem 6.3.
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The intersection graph of the entire set of 27 lines on a cubic surface is the complement of the Schläfli graph.
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Theory (B) 12 (1972), 32-40. attributed a graph with the mentioned properties of the Ljubljana graph to R. M. Foster.
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The 99-graph problem describes the smallest of these combinations of parameters for which the existence of a graph is unknown.
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In the mathematical field of graph theory, the Bidiakis cube is a 3-regular graph with 12 vertices and 18 edges.
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In extremal graph theory, the Erdős–Stone theorem is an asymptotic result generalising Turán's theorem to bound the number of edges in an H-free graph for a non-complete graph H. It is named after Paul Erdős and Arthur Stone, who proved it in 1946, and it has been described as the “fundamental theorem of extremal graph theory”.
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The Petersen family. K6 is at the top of the illustration, and the Petersen graph is at the bottom. The blue links indicate Δ-Y or Y-Δ transforms between graphs in the family. In graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph K6.
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The skeleton of the cube (the vertices and edges) form a graph, with 8 vertices, and 12 edges. It is a special case of the hypercube graph. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid. An extension is the three dimensional k-ary Hamming graph, which for k = 2 is the cube graph.
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From the properties of the zigzag product mentioned above, we see that the product of a large graph with a small graph, inherits a size similar to the large graph, and degree similar to the small graph, while preserving its expansion properties from both, thus enabling to increase the size of the expander without deleterious effects.
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Testing whether a graph is -tough is co-NP-complete. That is, the decision problem whose answer is "yes" for a graph that is not 1-tough, and "no" for a graph that is 1-tough, is NP-complete. The same is true for any fixed positive rational number : testing whether a graph is -tough is co-NP-complete .
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A circle graph, the intersection graph of chords of a circle. For book embeddings with a fixed vertex order, finding the book thickness is equivalent to coloring a derived circle graph. Finding the book thickness of a graph is NP-hard. This follows from the fact that finding Hamiltonian cycles in maximal planar graphs is NP-complete.
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The Graph API is the core of Facebook Platform, enabling developers to read from and write data into Facebook. The Graph API presents a simple, consistent view of the Facebook social graph, uniformly representing objects in the graph (e.g., people, photos, events, and pages) and the connections between them (e.g., friend relationships, shared content, and photo tags).
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In graph theory, an edge-graceful graph labeling is a type of graph labeling. This is a labeling for simple graphs in which no two distinct edges connect the same two distinct vertices, no edge connects a vertex to itself, and the graph is connected. Edge-graceful labelings were first introduced by Sheng- Ping Lo in his seminal paper.
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The automorphism group of the Dyck graph is a group of order 192.Royle, G. F032A data It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Dyck graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge.
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The Desargues graph has rectilinear crossing number 6, and is the smallest cubic graph with that crossing number . It is the only known nonplanar cubic partial cube. The Desargues graph has chromatic number 2, chromatic index 3, radius 5, diameter 5 and girth 6. It is also a 3-vertex-connected and a 3-edge-connected Hamiltonian graph.
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From 2012 he is a professor emeritus at the University of Kragujevac. His research interests are theoretical organic chemistry, physical chemistry, mathematical chemistry, graph theory, spectral graph theory and discrete mathematics. Gutman is known for his work in chemical graph theory and topological descriptors. In mathematics he introduced the notion of graph energy, a concept originating from theoretical chemistry.
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However, the hunt for the smallest planar hypohamiltonian graph continues. This question was first raised by Václav Chvátal in 1973. The answer is provided in 1976 by Carsten Thomassen, who exhibits a 105-vertices construction, the 105-Thomassen graph. In 1979, Hatzel improves this result with a planar hypohamiltonian graph on 57 vertices : the Hatzel graph.
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The Petersen graph is hypo-Hamiltonian: by deleting any vertex, such as the center vertex in the drawing, the remaining graph is Hamiltonian. This drawing with order-3 symmetry is the one given by . The Petersen graph has a Hamiltonian path but no Hamiltonian cycle. It is the smallest bridgeless cubic graph with no Hamiltonian cycle.
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The nine-vertex Paley graph is locally linear. One of its six triangles is highlighted in green. In graph theory, a locally linear graph is an undirected graph in which the neighborhood of every vertex is an induced matching. That is, for every vertex v, every neighbor of v should be adjacent to exactly one other neighbor of v.
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A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph., p. 6. In the abstract, all that matters is which pairs of vertices are connected by edges.
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A graph is (a,b)-decomposable if its edges can be partitioned into a+1 sets, each one of them inducing a forest, except one who induces a graph with maximum degree b. A graph with arboricity a is (a,0)-decomposable. The tree number is the minimal number of trees covering the edges of a graph.
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In graph theory, a connected graph is k-edge-connected if it remains connected whenever fewer than k edges are removed. The edge-connectivity of a graph is the largest k for which the graph is k-edge-connected. Edge connectivity and the enumeration of k-edge-connected graphs was studied by Camille Jordan in 1869.
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There are many synonyms for "cycle graph". These include simple cycle graph and cyclic graph, although the latter term is less often used, because it can also refer to graphs which are merely not acyclic. Among graph theorists, cycle, polygon, or n-gon are also often used. The term n-cycle is sometimes used in other settings.
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A graph and two of its cuts. The dotted line in red is a cut with three crossing edges. The dashed line in green is a min-cut of this graph, crossing only two edges. In computer science and graph theory, Karger's algorithm is a randomized algorithm to compute a minimum cut of a connected graph.
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The edge-connectivity of a vertex-transitive graph is equal to the degree d, while the vertex-connectivity will be at least 2(d + 1)/3. If the degree is 4 or less, or the graph is also edge-transitive, or the graph is a minimal Cayley graph, then the vertex-connectivity will also be equal to d.
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In the mathematical field of graph theory, the Shrikhande graph is a named graph discovered by S. S. Shrikhande in 1959.. It is a strongly regular graph with 16 vertices and 48 edges, with each vertex having degree 6\. Every pair of nodes has exactly two other neighbors in common, whether the pair of nodes is connected or not.
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In mathematics, the Perkel graph, named after Manley Perkel, is a 6-regular graph with 57 vertices and 171 edges. It is the unique distance-regular graph with intersection array (6, 5, 2; 1, 1, 3).Coolsaet, K. and Degraer, J. "A Computer Assisted Proof of the Uniqueness of the Perkel Graph." Designs, Codes and Crypt.
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It can also be described as the shortest path distance in a rotation graph, a graph that has a vertex for each binary tree on a given left-to-right sequence of nodes and an edge for each rotation between two trees. This rotation graph is exactly the graph of vertices and edges of an associahedron.
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The total graph T(G) of a graph G has as its vertices the elements (vertices or edges) of G, and has an edge between two elements whenever they are either incident or adjacent. The total graph may also be obtained by subdividing each edge of G and then taking the square of the subdivided graph., p. 82.
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A graph is a 2-leaf power if and only if it is the disjoint union of cliques (i.e., a cluster graph). A graph is a 3-leaf power if and only if it is a (bull, dart, gem)-free chordal graph.; Based on this characterization and similar ones, 3-leaf powers can be recognized in linear time.
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In graph theory, the modular decomposition is a decomposition of a graph into subsets of vertices called modules. A module is a generalization of a connected component of a graph. Unlike connected components, however, one module can be a proper subset of another. Modules therefore lead to a recursive (hierarchical) decomposition of the graph, instead of just a partition.
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Master Thesis, University of Tübingen, 2018 The characteristic polynomial of the Heawood graph is (x-3) (x+3) (x^2-2)^6. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
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The Dyck graph is the skeleton of a symmetric tessellation of a surface of genus three by twelve octagons, known as the Dyck map or Dyck tiling. The dual graph for this tiling is the complete tripartite graph K4,4,4...
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In chemistry, the Desargues graph is known as the Desargues–Levi graph; it is used to organize systems of stereoisomers of 5-ligand compounds. In this application, the thirty edges of the graph correspond to pseudorotations of the ligands.
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Of these, two are known to exist: the Berlekamp–van Lint–Seidel graph and the 9-vertex Paley graph with parameters (9,4,1,2). Conway's 99-graph problem concerns the existence of another of these graphs, the one with parameters (99,14,1,2).
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He was elected as a Fellow of the American Mathematical Society in the 2020 Class, for "contributions to topological graph theory, including the theory of graph embedding algorithms, graph coloring and crossing numbers, and for service to the profession".
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The inclusion exclusion principle forms the basis of algorithms for a number of NP-hard graph partitioning problems, such as graph coloring. A well known application of the principle is the construction of the chromatic polynomial of a graph.
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In the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices. Finding a matching in a bipartite graph can be treated as a network flow problem.
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Sumner's Universal Tournament Conjecture, Douglas B. West, retrieved 2010-09-17. A family of graphs has a universal graph of polynomial size, containing every -vertex graph as an induced subgraph, if and only if it has an adjacency labelling scheme in which vertices may be labeled by -bit bitstrings such that an algorithm can determine whether two vertices are adjacent by examining their labels. For, if a universal graph of this type exists, the vertices of any graph in may be labeled by the identities of the corresponding vertices in the universal graph, and conversely if a labeling scheme exists then a universal graph may be constructed having a vertex for every possible label.. In older mathematical terminology, the phrase "universal graph" was sometimes used to denote a complete graph. The notion of universal graph has been adapted and used for solving mean payoff games.
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Graph Modeling Language (GML) is a hierarchical ASCII-based file format for describing graphs. It has been also named Graph Meta Language.
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The graph with one node per 6-cycle, and one edge for each disjoint pair of 6-cycles, is the Coxeter graph..
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The order-5 folded cube graph (the 5-regular Clebsch graph) may be constructed by adding edges between opposite pairs of vertices in a 4-dimensional hypercube graph. (In an n-dimensional hypercube, a pair of vertices are opposite if the shortest path between them has n edges.) Alternatively, it can be formed from a 5-dimensional hypercube graph by identifying together (or contracting) every opposite pair of vertices. Another construction, leading to the same graph, is to create a vertex for each element of the finite field GF(16), and connect two vertices by an edge whenever the difference between the corresponding two field elements is a perfect cube. The order-5 halved cube graph (the 10-regular Clebsch graph) is the complement of the 5-regular graph.
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The Levi graph of the Desargues configuration, a graph having one vertex for each point or line in the configuration, is known as the Desargues graph. Because of the symmetries and self-duality of the Desargues configuration, the Desargues graph is a symmetric graph. The Petersen graph, in the layout shown by draws a different graph for this configuration, with ten vertices representing its ten lines, and with two vertices connected by an edge whenever the corresponding two lines do not meet at one of the points of the configuration. Alternatively, the vertices of this graph may be interpreted as representing the points of the Desargues configuration, in which case the edges connect pairs of points for which the line connecting them is not part of the configuration.
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A flow graph is a form of digraph associated with a set of linear algebraic or differential equations: :"A signal flow graph is a network of nodes (or points) interconnected by directed branches, representing a set of linear algebraic equations. The nodes in a flow graph are used to represent the variables, or parameters, and the connecting branches represent the coefficients relating these variables to one another. The flow graph is associated with a number of simple rules which enable every possible solution [related to the equations] to be obtained." Although this definition uses the terms "signal flow graph" and "flow graph" interchangeably, the term "signal flow graph" is most often used to designate the Mason signal-flow graph, Mason being the originator of this terminology in his work on electrical networks.
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A symmetric embedding of the Nauru graph on a genus-4 surface, with six dodecagonal faces. The Nauru graph has two different embeddings as a generalized regular polyhedron: a topological surface partitioned into edges, vertices, and faces in such a way that there is a symmetry taking any flag (an incident triple of a vertex, edge, and face) into any other flag.. One of these two embeddings forms a torus, so the Nauru graph is a toroidal graph: it consists of 12 hexagonal faces together with the 24 vertices and 36 edges of the Nauru graph. The dual graph of this embedding is a symmetric 6-regular graph with 12 vertices and 36 edges. The other symmetric embedding of the Nauru graph has six dodecagonal faces, and forms a surface of genus 4.
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The Moser spindle is a planar graph, meaning that it can be drawn without crossings in the plane. However, it is not possible to form such a drawing with straight line edges that is also a unit distance drawing; that is, it is not a matchstick graph. The Moser spindle is also a Laman graph, meaning that it forms a minimally rigid system when embedded in the plane. As a planar Laman graph, it is the graph of a pointed pseudotriangulation, meaning that it can be embedded in the plane in such a way that the unbounded face is the convex hull of the embedding and every bounded face is a pseudotriangle with only three convex vertices.. The complement graph of the Moser graph is a triangle-free graph.
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The existence of the Grötzsch graph demonstrates that the assumption of planarity is necessary in Grötzsch's theorem that every triangle-free planar graph is 3-colorable. used a modified version of the Grötzsch graph to disprove a conjecture of on the chromatic number of triangle-free graphs with high degree. Häggkvist's modification consists of replacing each of the five degree-four vertices of the Grötzsch graph by a set of three vertices, replacing each of the five degree-three vertices of the Grötzsch graph by a set of two vertices, and replacing the remaining degree-five vertex of the Grötzsch graph by a set of four vertices. Two vertices in this expanded graph are connected by an edge if they correspond to vertices connected by an edge in the Grötzsch graph.
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This construction is reversible; given a simple graph G, adjoin a new element x to the set of vertices of G and define the two-graph whose triples are all the {x, y, z} where y and z are adjacent vertices in G. This two-graph is called the extension of G by x in design theoretic language. In a given switching class of graphs of a regular two-graph, let Γx be the unique graph having x as an isolated vertex (this always exists, just take any graph in the class and switch the open neighborhood of x) without the vertex x. That is, the two-graph is the extension of Γx by x. In the first example above of a regular two-graph, Γx is a 5-cycle for any choice of x.
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The following figure illustrates the universal covering graph T of a graph H; the colours indicate the covering map. :File:Covering-graph-5.svg For any k, all k-regular graphs have the same universal cover: the infinite k-regular tree.
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In effect, a graph has two phases: connected (most nodes are linked by pathways of interaction) and fragmented (nodes are either isolated or form small subgraphs). These are often referred to as global and local phases, respectively. Fragmented graph. Connected graph.
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The triangle graph has chromatic number 3, chromatic index 3, radius 1, diameter 1 and girth 3. It is also a 2-vertex-connected graph and a 2-edge-connected graph. Its chromatic polynomial is (x-2)(x-1)x.
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The characteristic polynomial of the Nauru graph is equal to :(x-3) (x-2)^6 (x-1)^3 x^4 (x+1)^3 (x+2)^6 (x+3),\ making it an integral graph—a graph whose spectrum consists entirely of integers.
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The Harborth Graph. Harborth's research ranges across the subject areas of combinatorics, graph theory, discrete geometry, and number theory. In 1974, Harborth solved the unit coin graph problem,Heiko Harborth, Lösung zu Problem 664A, Elem. Math. 29 (1974), 14–15.
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When the configuration parameter t goes to infinity, shortest- path graph become the minimum spanning tree of the point set. The graph is a subgraph of the point set's Gabriel graph and therefore also a subgraph of its Delaunay triangulation.
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The definition of a composite service contains an implicit directed graph of inner service dependencies. The runtime environment for SOP can create an execution graph based on this directed graph by automatically instantiating and running inner services in parallel whenever possible.
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In statistics and Markov modeling, an ancestral graph is a type of mixed graph to provide a graphical representation for the result of marginalizing one or more vertices in a graphical model that takes the form of a directed acyclic graph.
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The different types of edge in a bidirected graph In the mathematical domain of graph theory, a bidirected graph (introduced by ). Reprinted in Combinatorial Optimization — Eureka, You Shrink!, Springer-Verlag, Lecture Notes in Computer Science 2570, 2003, pp. 27–30, .
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The halved cube graph of order 4, obtained as the bipartite half of an order-4 hypercube graph In graph theory, the bipartite half or half-square of a bipartite graph G = (U,V,E) is a graph whose vertex set is one of the two sides of the bipartition (without loss of generality, U) and in which there is an edge uiuj for each two vertices ui and uj in U that are at distance two from each other in G.. That is, in a more compact notation, the bipartite half is G2[U] where the superscript 2 denotes the square of a graph and the square brackets denote an induced subgraph. For instance, the bipartite half of the complete bipartite graph Kn,n is the complete graph Kn and the bipartite half of the hypercube graph is the halved cube graph. When G is a distance-regular graph, its two bipartite halves are both distance-regular.. For instance, the halved Foster graph is one of finitely many degree-6 distance-regular locally linear graphs. The map graphs, that is, the intersection graphs of interior- disjoint simply-connected regions in the plane, are exactly the bipartite halves of bipartite planar graphs..
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In any directed bipartite graph, all cycles have a length that is divisible by two. Therefore, no directed bipartite graph can be aperiodic. In any directed acyclic graph, it is a vacuous truth that every k divides all cycles (because there are no directed cycles to divide) so no directed acyclic graph can be aperiodic. And in any directed cycle graph, there is only one cycle, so every cycle's length is divisible by n, the length of that cycle.
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Examples of existing extensions include the ones for directed graphs, 3D graph drawing, cluster graph drawing, constrained graph drawing, and dynamic graph drawing. ; Intuitive: Since they are based on physical analogies of common objects, like springs, the behavior of the algorithms is relatively easy to predict and understand. This is not the case with other types of graph-drawing algorithms. ; Simplicity: Typical force-directed algorithms are simple and can be implemented in a few lines of code.
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The Wagner graph is a vertex-transitive graph but is not edge-transitive. Its full automorphism group is isomorphic to the dihedral group D8 of order 16, the group of symmetries of an octagon, including both rotations and reflections. The characteristic polynomial of the Wagner graph is (x-3)(x-1)^2(x+1)(x^2+2x-1)^2. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
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Edge coloring Tietze's graph requires four colors; that is, its chromatic index is 4. Equivalently, the edges of Tietze's graph can be partitioned into four matchings, but no fewer. Tietze's graph matches part of the definition of a snark: it is a cubic bridgeless graph that is not 3-edge- colorable. However, some authors restrict snarks to graphs without 3-cycles and 4-cycles, and under this more restrictive definition Tietze's graph is not a snark.
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The complete graph on vertices is denoted by . Some sources claim that the letter K in this notation stands for the German word komplett,. but the German name for a complete graph, vollständiger Graph, does not contain the letter K, and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory.. Kn has edges (a triangular number), and is a regular graph of degree . All complete graphs are their own maximal cliques.
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On a Sudoku board of size n^2\times n^2, the Sudoku graph has n^4 vertices, each with exactly 3n^2-2n-1 neighbors. Therefore, it is a regular graph. For instance, the graph shown in the figure, for a 4\times 4 board, has 16 vertices and is 7-regular. For the most common form of Sudoku, on a 9\times 9 board, the Sudoku graph is a 20-regular graph with 81 vertices.
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Each edge in this drawing is crossed at most once, so the Petersen graph is 1-planar. On a torus the Petersen graph can be drawn without edge crossings; it therefore has orientable genus 1. The Petersen graph is a unit distance graph: it can be drawn in the plane with each edge having unit length. The Petersen graph can also be drawn (with crossings) in the plane in such a way that all the edges have equal length.
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A graph with three vertices and three edges. A graph (sometimes called undirected graph for distinguishing from a directed graph, or simple graph for distinguishing from a multigraph)See, for instance, Iyanaga and Kawada, 69 J, p. 234 or Biggs, p. 4. is a pair , where is a set whose elements are called vertices (singular: vertex), and is a set of two-sets (sets with two distinct elements) of vertices, whose elements are called edges (sometimes links or lines).
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The graph coloring game is a mathematical game related to graph theory. Coloring game problems arose as game-theoretic versions of well-known graph coloring problems. In a coloring game, two players use a given set of colors to construct a coloring of a graph, following specific rules depending on the game we consider. One player tries to successfully complete the coloring of the graph, when the other one tries to prevent him from achieving it.
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In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph K3,3 is planar.. Bóna introduces the puzzle (in the form of three houses to be connected to three wells) on p. 275, and writes on p. 277 that it "is equivalent to the problem of drawing K3,3 on a plane surface without crossings". This graph is often referred to as the utility graph in reference to the problem;Utility Graph from mathworld.wolfram.
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However, the same definitions apply to directed graphs and a directed graph is also equivalent to a unique core. Every graph and every directed graph contains its core as a retract and as an induced subgraph. For example, all complete graphs Kn and all odd cycles (cycle graphs of odd length) are cores. Every 3-colorable graph G that contains a triangle (that is, has the complete graph K3 as a subgraph) is homomorphically equivalent to K3.
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The bipartite double cover of any graph G is a bipartite graph; both parts of the bipartite graph have one vertex for each vertex of G. A bipartite double cover is connected if and only if G is connected and non-bipartite., Theorem 3.4. The bipartite double cover is a special case of a double cover (a 2-fold covering graph). A double cover in graph theory can be viewed as a special case of a topological double cover.
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A DPO graph transformation system (or graph grammar) consists of a finite graph, which is the starting state, and a finite or countable set of labeled spans in the category of finite graphs and graph homomorphisms, which serve as derivation rules. The rule spans are generally taken to be composed of monomorphisms, but the details can vary."Double-pushout graph transformation revisited", Habel, Annegret and Müller, Jürgen and Plump, Detlef, Mathematical Structures in Computer Science, vol. 11, no. 05.
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A graph with odd-crossing number 13 and pair-crossing number 15. In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and the edges by Jordan arcs (connected pieces of Jordan curves) joining the corresponding pairs of points. The points representing the vertices of a graph and the arcs representing its edges are called the vertices and the edges of the topological graph. It is usually assumed that any two edges of a topological graph cross a finite number of times, no edge passes through a vertex different from its endpoints, and no two edges touch each other (without crossing).
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It is possible to determine whether a graph is strongly chordal in polynomial time, by repeatedly searching for and removing a simple vertex. If this process eliminates all vertices in the graph, the graph must be strongly chordal; otherwise, if this process finds a subgraph without any more simple vertices, the original graph cannot be strongly chordal. For a strongly chordal graph, the order in which the vertices are removed by this process is a strong perfect elimination ordering.. Alternative algorithms are now known that can determine whether a graph is strongly chordal and, if so, construct a strong perfect elimination ordering more efficiently, in time for a graph with n vertices and m edges.; ; .
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The Moser spindle, a planar Laman graph drawn as a pointed pseudotriangulation The complete bipartite graph K3,3, a non-planar Laman graph In graph theory, the Laman graphs are a family of sparse graphs describing the minimally rigid systems of rods and joints in the plane. Formally, a Laman graph is a graph on n vertices such that, for all k, every k-vertex subgraph has at most 2k − 3 edges, and such that the whole graph has exactly 2n − 3 edges. Laman graphs are named after Gerard Laman, of the University of Amsterdam, who in 1970 used them to characterize rigid planar structures.. This characterization, however, had already been discovered in 1927 by Hilda Geiringer..
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They can be used to define the tree-depth of a graph, and as part of the left-right planarity test for testing whether a graph is a planar graph. A characterization of Trémaux trees in the monadic second-order logic of graphs allows graph properties involving orientations to be recognized efficiently for graphs of bounded treewidth using Courcelle's theorem. Not every infinite connected graph has a Trémaux tree, and the graphs that do have them can be characterized by their forbidden minors. A Trémaux tree exists in every connected graph with countably many vertices, even when an infinite form of depth-first search would not succeed in exploring every vertex of the graph.
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This can be proved by showing how to convert an arbitrary graph into a planar graph, such that a game of GG played on this graph will have the same outcome as on the original graph. In order to do that, it's only necessary to eliminate all the edge crossings of the original graph. We draw the graph such that no three edges intersect at a point, and no pair of crossing edges can both be used in the same game. This is not possible in general, but is always possible for the graph constructed from a FORMULA-GAME instance; for example we could have only the edges from clause vertices involved in crossings.
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A perfect matching (red edges), in the Petersen graph. Since the Petersen graph is cubic and bridgeless, it meets the conditions of Petersen's theorem. A cubic (but not bridgeless) graph with no perfect matching, showing that the bridgeless condition in Petersen's theorem cannot be omitted In the mathematical discipline of graph theory, Petersen's theorem, named after Julius Petersen, is one of the earliest results in graph theory and can be stated as follows: > Petersen's Theorem. Every cubic, bridgeless graph contains a perfect > matching.. In other words, if a graph has exactly three edges at each vertex, and every edge belongs to a cycle, then it has a set of edges that touches every vertex exactly once.
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An undirected graph G is Hamiltonian if it contains a cycle that touches each of its vertices exactly once. It is 2-vertex-connected if it does not have an articulation vertex, a vertex whose deletion would leave the remaining graph disconnected. Not every 2-vertex- connected graph is Hamiltonian; counterexamples include the Petersen graph and the complete bipartite graph K2,3. The square of G is a graph G2 that has the same vertex set as G, and in which two vertices are adjacent if and only if they have distance at most two in G. Fleischner's theorem states that the square of a finite 2-vertex-connected graph with at least three vertices must always be Hamiltonian.
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A first order sentence in the logic of graphs is said to define a graph if is the only graph that models . Every graph may be defined by at least one sentence; for instance, one can define an -vertex graph by a sentence with variables, one for each vertex of the graph, and one more to state the condition that there is no vertex other than the vertices of the graph. Additional clauses of the sentence can be used to ensure that no two vertex variables are equal, that each edge of is present, and no edge exists between a pair of non-adjacent vertices of . However, for some graphs there exist significantly shorter formulas that define the graph.
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In computational geometry, the Theta graph, or \Theta-graph, is a type of geometric spanner similar to a Yao graph. The basic method of construction involves partitioning the space around each vertex into a set of cones, which themselves partition the remaining vertices of the graph. Like Yao Graphs, a \Theta-graph contains at most one edge per cone; where they differ is how that edge is selected. Whereas Yao Graphs will select the nearest vertex according to the metric space of the graph, the \Theta-graph defines a fixed ray contained within each cone (conventionally the bisector of the cone) and selects the nearest neighbor with respect to orthogonal projections to that ray.
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A cut vertex is a vertex the removal of which would disconnect the remaining graph; a vertex separator is a collection of vertices the removal of which would disconnect the remaining graph into small pieces. A k-vertex-connected graph is a graph in which removing fewer than k vertices always leaves the remaining graph connected. An independent set is a set of vertices no two of which are adjacent, and a vertex cover is a set of vertices that includes at least one endpoint of each edge in the graph. The vertex space of a graph is a vector space having a set of basis vectors corresponding with the graph's vertices.
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If we now perform the same type of random walk on the vertices of the line graph, the frequency with which v is visited can be completely different from f. If our edge e in G was connected to nodes of degree O(k), it will be traversed O(k2) more frequently in the line graph L(G). Put another way, the Whitney graph isomorphism theorem guarantees that the line graph almost always encodes the topology of the original graph G faithfully but it does not guarantee that dynamics on these two graphs have a simple relationship. One solution is to construct a weighted line graph, that is, a line graph with weighted edges.
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A penny graph with 11 vertices and 19 edges that requires four colors in any graph coloring A four-coloring of the graph above. In geometric graph theory, a penny graph is a contact graph of unit circles. That is, it is an undirected graph whose vertices can be represented by unit circles, with no two of these circles crossing each other, and with two adjacent vertices if and only if they are represented by tangent circles.. See especially p. 176. More simply, they are the graphs formed by arranging pennies in a non-overlapping way on a flat surface, making a vertex for each penny, and making an edge for each two pennies that touch.
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In mathematics, a two-graph is a set of (unordered) triples chosen from a finite vertex set X, such that every (unordered) quadruple from X contains an even number of triples of the two-graph. A regular two-graph has the property that every pair of vertices lies in the same number of triples of the two- graph. Two-graphs have been studied because of their connection with equiangular lines and, for regular two-graphs, strongly regular graphs, and also finite groups because many regular two-graphs have interesting automorphism groups. A two-graph is not a graph and should not be confused with other objects called 2-graphs in graph theory, such as 2-regular graphs.
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In graph theory, the term bipartite hypergraph describes several related classes of hypergraphs, all of which are natural generalizations of a bipartite graph.
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The assembly graph is filtered and untangled, producing all possible paths of the graph, and therefore all configurations of the circular organellar genomes.
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A directed acyclic graph G has an upward planar drawing if and only if G is a subgraph of an st-planar graph..
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There are two kinds of matroid associated with a biased graph, both of which generalize the cycle matroid of a graph (Zaslavsky, 1991).
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In graph theory, a connected dominating set and a maximum leaf spanning tree are two closely related structures defined on an undirected graph.
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However, unlike the Strahler number, the pathwidth is defined only for the whole graph, and not separately for each node in the graph.
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Especially when used on heuristics considering the graph only locally, as the multi-level technique constitutes a more global view on the graph.
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An incomparability graph is an undirected graph that connects pairs of elements that are not comparable to each other in a partial order.
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Every cluster graph is a block graph, a cograph, and a claw-free graph. Every maximal independent set in a cluster graph chooses a single vertex from each cluster, so the size of such a set always equals the number of clusters; because all maximal independent sets have the same size, cluster graphs are well-covered. The Turán graphs are complement graphs of cluster graphs, with all complete subgraphs of equal or nearly-equal size. The locally clustered graph (graphs in which every neighborhood is a cluster graph) are the diamond-free graphs, another family of graphs that contains the cluster graphs.
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A graph with 16 vertices and 6 bridges (highlighted in red) An undirected connected graph with no bridge edges In graph theory, a bridge, isthmus, cut- edge, or cut arc is an edge of a graph whose deletion increases the graph's number of connected components.. Equivalently, an edge is a bridge if and only if it is not contained in any cycle. For a connected graph, a bridge can uniquely determine a cut. A graph is said to be bridgeless or isthmus-free if it contains no bridges. Another meaning of "bridge" appears in the term bridge of a subgraph.
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An example of planar straight-line graph In computational geometry, a planar straight-line graph, in short PSLG, (or straight-line plane graph, or plane straight-line graph) is a term used for an embedding of a planar graph in the plane such that its edges are mapped into straight line segments. Fáry's theorem (1948) states that every planar graph has this kind of embedding. In computational geometry, PSLGs have often been called planar subdivisions, with an assumption or assertion that subdivisions are polygonal rather than having curved boundaries. PSLGs may serve as representations of various maps, e.g.
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If a walk-regular graph admits perfect state transfer, then all of its eigenvalues are integers. If G is a graph in a homogeneous coherent algebra that admits perfect state transfer at time t, such as e.g. a vertex-transitive graph or a graph in an association scheme, then all of the vertices on G admit perfect state transfer at time t. Moreover, a graph G must have a perfect matching that admits perfect state transfer if it admits perfect state transfer between a pair of adjacent vertices and is a graph in a homogeneous coherent algebra.
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In the mathematical field of graph theory, the Robertson–Wegner graph is a 5-regular undirected graph with 30 vertices and 75 edges named after Neil Robertson and G. Wegner.Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 238, 1976.Wong, P. K. "A note on a paper of G. Wegner", Journal of Combinatorial Theory, Series B, 22:3, June 1977, pgs 302-303, doi:10.1016/0095-8956(77)90081-8 It is one of the four (5,5)-cage graphs, the others being the Foster cage, the Meringer graph, and the Wong graph.
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The smallest cubic semi-symmetric graph (that is, one in which each vertex is incident to exactly three edges) is the Gray graph on 54 vertices. It was first observed to be semi-symmetric by . It was proven to be the smallest cubic semi-symmetric graph by Dragan Marušič and Aleksander Malnič.. All the cubic semi-symmetric graphs on up to 768 vertices are known. According to Conder, Malnič, Marušič and Potočnik, the four smallest possible cubic semi-symmetric graphs after the Gray graph are the Iofinova-Ivanov graph on 110 vertices, the Ljubljana graph on 112 vertices,.
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A three-page book embedding of the complete graph . Because it is not a planar graph, it is not possible to embed this graph without crossings on fewer pages, so its book thickness is three. In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings into a book, a collection of half-planes all having the same line as their boundary. Usually, the vertices of the graph are required to lie on this boundary line, called the spine, and the edges are required to stay within a single half-plane.
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The only other possible set of parameters yielding a Moore graph is (3250, 57, 0, 1); it is unknown if such a graph exists, and if so, whether or not it is unique. More generally, every strongly regular graph with \mu=1 is a geodetic graph, a graph in which every two vertices have a unique unweighted shortest path. The only known strongly regular graphs with \mu=1 are the Moore graphs. It is not possible for such a graph to have \lambda=1, but other combinations of parameters such as (400, 21, 2, 1) have not yet been ruled out.
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There is no natural concept of distance (a metric) in an incidence structure. However, a combinatorial metric does exist in the corresponding incidence graph (Levi graph), namely the length of the shortest path between two vertices in this bipartite graph. The distance between two objects of an incidence structure – two points, two lines or a point and a line – can be defined to be the distance between the corresponding vertices in the incidence graph of the incidence structure. Another way to define a distance again uses a graph-theoretic notion in a related structure, this time the collinearity graph of the incidence structure.
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The skeleton of any convex polyhedron is a planar graph, and the skeleton of any k-dimensional convex polytope is a k-connected graph. Conversely, Steinitz's theorem states that any 3-connected planar graph is the skeleton of a convex polyhedron; for this reason, this class of graphs is also known as the polyhedral graphs. A Euclidean graph is a graph in which the vertices represent points in the plane, and the edges are assigned lengths equal to the Euclidean distance between those points. The Euclidean minimum spanning tree is the minimum spanning tree of a Euclidean complete graph.
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The line graph of the Petersen graph, another graph of this type, has a property analogous to the cages in that it is the smallest possible graph in which the largest clique has three vertices, each vertex is in exactly two edge-disjoint cliques, and the shortest cycle with edges from distinct cliques has length five. A more complicated expansion process applies to planar graphs. Let G be a planar graph embedded in the plane in such a way that every face is a quadrilateral, such as the graph of a cube. Necessarily, if G has n vertices, it has n-2 faces.
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The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of forbidden graphs, now known as Kuratowski's theorem: :A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph K5 or the complete bipartite graph K3,3 (utility graph). A subdivision of a graph results from inserting vertices into edges (for example, changing an edge •----• to •--•--•) zero or more times. An example of a graph with no K5 or K3,3 subgraph. However, it contains a subdivision of K3,3 and is therefore non-planar.
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Witsenhausen (1974) conjectures that the maximum sum of squared distances, among n points with unit diameter in Rd, is attained for a configuration formed by embedding a Turán graph onto the vertices of a regular simplex. An n-vertex graph G is a subgraph of a Turán graph T(n,r) if and only if G admits an equitable coloring with r colors. The partition of the Turán graph into independent sets corresponds to the partition of G into color classes. In particular, the Turán graph is the unique maximal n-vertex graph with an r-color equitable coloring.
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In the mathematical field of graph theory, the Ellingham–Horton graphs are two 3-regular graphs on 54 and 78 vertices: the Ellingham–Horton 54-graph and the Ellingham–Horton 78-graph. They are named after Joseph D. Horton and Mark N. Ellingham, their discoverers. These two graphs provide counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite graph is Hamiltonian.. The book thickness of the Ellingham-Horton 54-graph and the Ellingham-Horton 78-graph is 3 and the queue numbers 2Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, Universität Tübingen, 2018.
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The Petersen graph The Petersen graph is an undirected graph with ten vertices and fifteen edges, commonly drawn as a pentagram within a pentagon, with corresponding vertices attached to each other. It has many unusual mathematical properties, and has frequently been used as a counterexample to conjectures in graph theory. The book uses these properties as an excuse to cover several advanced topics in graph theory where this graph plays an important role. It is heavily illustrated, and includes both open problems on the topics it discusses and detailed references to the literature on these problems.
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Such an orientation can be found by starting with an arbitrary orientation of a spanning tree of the graph. The remaining edges, not in this tree, form a spanning tree of the dual graph, and their orientations can be chosen according to a bottom-up traversal of the dual spanning tree in order to ensure that each face of the original graph has an odd number of clockwise edges. More generally, every K_{3,3}-minor-free graph has a Pfaffian orientation. These are the graphs that do not have the utility graph K_{3,3} (which is not Pfaffian) as a graph minor.
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There is an even more symmetric embedding of Möbius–Kantor graph in the double torus which is a regular map, with six octagonal faces, in which all 96 symmetries of the graph can be realized as symmetries of the embedding; credits this embedding to . Its 96-element symmetry group has a Cayley graph that can itself be embedded on the double torus, and was shown by to be the unique group with genus two. The Cayley graph on 96 vertices is a flag graph of the genus 2 regular map having Möbius–Kantor graph as a skeleton.
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A graph with edges colored to illustrate path H-A-B (green), closed path or walk with a repeated vertex B-D-E-F-D-C-B (blue) and a cycle with no repeated edge or vertex H-D-G-H (red). In graph theory, a cycle in a graph is a non- empty trail in which the only repeated vertices are the first and last vertices. A directed cycle in a directed graph is a non-empty directed trail in which the only repeated vertices are the first and last vertices. A graph without cycles is called an acyclic graph.
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The Wagner graph, an eight-vertex Möbius ladder arising in Wagner's characterization of K5-free graphs. Wagner is known for his contributions to graph theory and particularly the theory of graph minors, graphs that can be formed from a larger graph by contracting and removing edges. Wagner's theorem characterizes the planar graphs as exactly those graphs that do not have as a minor either a complete graph K5 on five vertices or a complete bipartite graph K3,3 with three vertices on each side of its bipartition. That is, these two graphs are the only minor-minimal non-planar graphs.
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Her work initiated a geometric approach to spectral graph theory with connections to differential geometry. According to the biography Fan Rong K Chung Graham, "Spectral graph theory studies how the spectrum of the Laplacian of a graph is related to its combinatorial properties.". In 1997, the American Mathematical Society published Chung's book Spectral graph theory. This book became a standard textbook at many universities and is the key to study Spectral graph theory for many mathematics students who are interested in this area. Fan Chung's study in the spectral graph theory brings this “algebraic connectivity” of graphs into a new and higher level.
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The mapping is called an isomorphism between and . When and there exists an isomorphism between the sub-graph and a graph , this mapping represents an appearance of in . The number of appearances of graph in is called the frequency of in . A graph is called recurrent (or frequent) in when its frequency is above a predefined threshold or cut-off value.
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Contracting the edge between the indicated vertices, resulting in graph G / {uv}. In graph theory, an edge contraction is an operation which removes an edge from a graph while simultaneously merging the two vertices that it previously joined. Edge contraction is a fundamental operation in the theory of graph minors. Vertex identification is a less restrictive form of this operation.
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A call graph generated for a simple computer program in Python. A call graph (also known as a call multigraph) is a control flow graph, which represents calling relationships between subroutines in a computer program. Each node represents a procedure and each edge (f, g) indicates that procedure f calls procedure g. Thus, a cycle in the graph indicates recursive procedure calls.
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Call graphs can be dynamic or static. A dynamic call graph is a record of an execution of the program, for example as output by a profiler. Thus, a dynamic call graph can be exact, but only describes one run of the program. A static call graph is a call graph intended to represent every possible run of the program.
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A graph G itself is said to be periodic if there is a time t eq 0 such that all of its vertices are periodic at time t. A graph is periodic if and only if its (non- zero) eigenvalues are all rational multiples of each other. Moreover, a regular graph is periodic if and only if it is an integral graph.
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The automorphism group of the Heawood graph is isomorphic to the projective linear group PGL2(7), a group of order 336. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Heawood graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge.
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The goal of detection algorithms is simply to determine, given a sampled graph, whether the graph has latent community structure. More precisely, a graph might be generated, with some known prior probability, from a known stochastic block model, and otherwise from a similar Erdos-Renyi model. The algorithmic task is to correctly identify which of these two underlying models generated the graph.
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The left multiplication action of any group on itself is simply transitive, in particular, the Cayley graph is vertex transitive. This leads to the following characterization of Cayley graphs: :Sabidussi's Theorem. A graph \Gamma is a Cayley graph of a group G if and only if it admits a simply transitive action of G by graph automorphisms (i.e. preserving the set of edges).
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In the mathematical field of graph theory, the Ljubljana graph is an undirected bipartite graph with 112 vertices and 168 edges. It is a cubic graph with diameter 8, radius 7, chromatic number 2 and chromatic index 3. Its girth is 10 and there are exactly 168 cycles of length 10 in it. There are also 168 cycles of length 12.
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The Barnette–Bosák–Lederberg graph has a similar construction to the Tutte graph but is composed of two Tutte fragments, connected through a pentagonal prism, instead of three connected through a tetrahedron. Without the constraint of having exactly three edges at every vertex, much smaller non-Hamiltonian polyhedral graphs are possible, including the Goldner–Harary graph and the Herschel graph.
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The automorphism group of the Möbius–Kantor graph is a group of order 96.Royle, G. F016A data It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Möbius–Kantor graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge.
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A useful graph that is often associated with a triangulation of a polygon is the dual graph. Given a triangulation of , one defines the graph as the graph whose vertex set are the triangles of , two vertices (triangles) being adjacent if and only if they share a diagonal. It is easy to observe that is a tree with maximum degree 3.
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On the left-top a vertex critical graph with chromatic number 6; next all the N-1 subgraphs with chromatic number 5. In graph theory, a critical graph is a graph G in which every vertex or edge is a critical element, that is, if its deletion decreases the chromatic number of G. Such a decrease cannot be by more than 1.
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The Journal of Graph Algorithms and Applications is an open access peer- reviewed scientific journal covering the subject of graph algorithms and graph drawing. The journal was established in 1997 and the editor-in-chief is Giuseppe Liotta (University of Perugia). It is abstracted and indexed by Scopus and MathSciNet.Journal Information for "Journal of Graph Algorithms and Applications", MathSciNet, retrieved 2011-03-02.
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The Games graph is a strongly regular graph with 729 vertices. Every edge belongs to a unique triangle, so it is a locally linear graph, the largest known locally linear strongly regular graph. Its construction is based on the unique 56-point cap set in the five-dimensional ternary projective space (rather than the affine space that cap-sets are commonly defined in)..
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The coloring of the graph may then be recovered by composing this homomorphism with the homomorphism from this tensor product to its K3 factor. However, the Clebsch graph and its tensor product with K3 are both non-planar; there does not exist a triangle-free planar graph to which every other triangle-free planar graph may be mapped by a homomorphism., Theorem 11; .
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The full automorphism group of the diamond graph is a group of order 4 isomorphic to the Klein four-group, the direct product of the cyclic group Z/2Z with itself. The characteristic polynomial of the diamond graph is x(x+1)(x^2-x-4). It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
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The problem of finding non-self-intersecting smooth curves between given points in a train track comes up in testing whether certain kinds of graph drawings are valid and may be modeled as the search for a regular path in a skew-symmetric graph. A closely related concept is the bidirected graph of ("polarized graph" in the terminology of , ), a graph in which each of the two ends of each edge may be either a head or a tail, independently of the other end. A bidirected graph may be interpreted as a polar graph by letting the partition of edges at each vertex be determined by the partition of endpoints at that vertex into heads and tails; however, swapping the roles of heads and tails at a single vertex ("switching" the vertex, in the terminology of ) produces a different bidirected graph but the same polar graph. For the correspondence between bidirected graphs and skew- symmetric graphs (i.e.
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The smallest 3-regular matchstick graph is formed from two copies of the diamond graph placed in such a way that corresponding vertices are at unit distance from each other; its bipartite double cover is the 8-crossed prism graph. In 1986, Heiko Harborth presented the graph that would bear his name, the Harborth Graph. With 104 edges and 52 vertices, is the smallest known example of a 4-regular matchstick graph.. As cited in: It is a rigid graph.. For additional details see Gerbracht's earlier preprint "Minimal Polynomials for the Coordinates of the Harborth Graph" (2006), arXiv:math/0609360. Every 4-regular matchstick graph contains at least 20 vertices.. Examples of 4-regular matchstick graphs are currently known for all number of vertices ≥ 52 except for 53, 55, 56, 58, 59, 61 and 62. The graphs with 54, 57, 65, 67, 73, 74, 77 and 85 vertices were first published in 2016.
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The symmetric group on five points is also the symmetry group of the Petersen graph, and the order-2 subgroup swaps the vertices within each pair of vertices formed in the double cover construction. The generalized Petersen graph G(n, k) is vertex- transitive if and only if n = 10 and k = 2 or if k2 ≡ ±1 (mod n) and is edge- transitive only in the following seven cases: (n, k) = (4, 1), (5, 2), (8, 3), (10, 2), (10, 3), (12, 5), (24, 5).. So the Desargues graph is one of only seven symmetric Generalized Petersen graphs. Among these seven graphs are the cubical graph G(4, 1), the Petersen graph G(5, 2), the Möbius–Kantor graph G(8, 3), the dodecahedral graph G(10, 2) and the Nauru graph G(12, 5). The characteristic polynomial of the Desargues graph is : (x-3) (x-2)^4 (x-1)^5 (x+1)^5 (x+2)^4 (x+3).
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If the algebra has the property that every interval is finite, then this graph is a median graph, and it accurately represents the algebra in that the median operation defined by shortest paths on the graph coincides with the algebra's original median operation.
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In graph theory, a branch of mathematics, the cop number or copnumber of an undirected graph is the minimum number of cops that suffices to ensure a win (i.e., a capture of the robber) in a certain pursuit-evasion game on the graph.
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In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced in , and has seen many uses in the field of universal algebra since then.
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In 1972, A. T. Balaban published a (3-10)-cage graph, a cubic graph that has as few vertices as possible for girth 10.A. T. Balaban, A trivalent graph of girth ten, J. Combin. Theory Ser. B 12, 1-5. 1972.
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M. O'Keefe and P.K. Wong, A smallest graph of girth 10 and valency 3, J. Combin. Theory Ser. B 29 (1980) 91-105. There exist three distinct (3-10)-cage graphs—the Balaban 10-cage, the Harries graph and the Harries–Wong graph.
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In 1972, A. T. Balaban published a (3-10)-cage graph, a cubic graph that has as few vertices as possible for girth 10.A. T. Balaban, A trivalent graph of girth ten, J. Combin. Theory Ser. B 12, 1-5\. 1972.
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M. O'Keefe and P.K. Wong, A smallest graph of girth 10 and valency 3, J. Combin. Theory Ser. B 29 (1980) 91-105. There exist three distinct (3-10)-cage graphs—the Balaban 10-cage, the Harries graph and the Harries-Wong graph.
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In the case of a productive capacity graph, on the horizontal line are defined capital goods and on the vertical line, consumer goods are stated. The functioning of the productive capacity graph is the same as for the above-mentioned PPF graph.
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Like the Rado graph, contains a bidirectional Hamiltonian path such that any symmetry of the path is a symmetry of the whole graph. However, this is not true for when : for these graphs, every automorphism of the graph has more than one orbit.
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An interval graph is called p-improper if there is a representation in which no interval contains more than p others. This notion extends the idea of proper interval graphs such that a 0-improper interval graph is a proper interval graph.
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The characteristic polynomial of the Chvátal graph is (x-4) (x-1)^4 x^2 (x+1) (x+3)^2 (x^2+x-4). The Tutte polynomial of the Chvátal graph has been computed by . The independence number of this graph is 4.
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In the mathematical field of graph theory, the Errera graph is a graph with 17 vertices and 45 edges. Alfred Errera published it in 1921 as a counterexample to Kempe's erroneous proof of the four color theorem; it was named after Errera by .
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In the mathematical area of graph theory, a graph is even-hole-free if it contains no induced cycle with an even number of vertices. demonstrated that every even-hole-free graph contains a bisimplicial vertex, which settled a conjecture by Reed.
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A very similar construction from PG(3,2) is used to build the Higman-Sims graph, which has the Hoffman-Singleton graph as a subgraph.
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For instance, the SPQR tree of a biconnected graph is a representation of the graph as a 2-clique-sum of its triconnected components.
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A planar graph is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect.
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The two most common measures of triadic closure for a graph are (in no particular order) the clustering coefficient and transitivity for that graph.
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Grochow and Kellis proposed an exact algorithm for enumerating sub-graph appearances. The algorithm is based on a motif-centric approach, which means that the frequency of a given sub-graph,called the query graph, is exhaustively determined by searching for all possible mappings from the query graph into the larger network. It is claimed that a motif-centric method in comparison to network-centric methods has some beneficial features. First of all it avoids the increased complexity of sub-graph enumeration.
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7, August 1978, pp. 219-222. In 1980 he introduced the relative neighborhood graph (RNG) to the fields of pattern recognition and machine learning, and showed that it contained the minimum spanning tree, and was a subgraph of the Delaunay triangulation. Three other well known proximity graphs are the nearest neighbor graph, the Urquhart graph, and the Gabriel graph. The first is contained in the minimum spanning tree, and the Urquhart graph contains the RNG, and is contained in the Delaunay triangulation.
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In graph theory, the Krackhardt kite graph is a simple graph with ten nodes. The graph is named after David Krackhardt, a researcher of social network theory. Krackhardt introduced the graph in 1990 to distinguish different concepts of centrality. It has the property that the vertex with maximum degree (labeled 3 in the figure, with degree 6), the vertex with maximum betweenness centrality (labeled 7), and the two vertices with maximum closeness centrality (labeled 5 and 6) are all different from each other.
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A rigid graph is an embedding of a graph in a Euclidean space which is structurally rigid. That is, a graph is rigid if the structure formed by replacing the edges by rigid rods and the vertices by flexible hinges is rigid. A graph that is not rigid is called flexible. More formally, a graph embedding is flexible if the vertices can be moved continuously, preserving the distances between adjacent vertices, with the result that the distances between some nonadjacent vertices are altered.
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The Rado graph arises almost surely in the Erdős–Rényi model of a random graph on countably many vertices. Specifically, one may form an infinite graph by choosing, independently and with probability 1/2 for each pair of vertices, whether to connect the two vertices by an edge. With probability 1 the resulting graph is isomorphic to the Rado graph. This construction also works if any fixed probability p not equal to 0 or 1 is used in place of 1/2.
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Reply by Urquhart, pp. 860–861. it can be used as a good approximation to it.. The problem of constructing relative neighborhood graphs in O(n log n) time, left open by the mismatch between the Urquhart graph and the relative neighborhood graph, was solved by .. Like the relative neighborhood graph, the Urquhart graph of a set of points in general position contains the Euclidean minimum spanning tree of its points, from which it follows that it is a connected graph.
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The genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n handles (i.e. an oriented surface of genus n). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps (i.e.
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In the mathematical field of graph theory, a convex bipartite graph is a bipartite graph with specific properties. A bipartite graph, (U ∪ V, E), is said to be convex over the vertex set U if U can be enumerated such that for all v ∈ V the vertices adjacent to v are consecutive. Convexity over V is defined analogously. A bipartite graph (U ∪ V, E) that is convex over both U and V is said to be biconvex or doubly convex.
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Bipartite Heawood graph. Points are represented by vertices of one color and lines by vertices of the other color. As with any incidence structure, the Levi graph of the Fano plane is a bipartite graph, the vertices of one part representing the points and the other representing the lines, with two vertices joined if the corresponding point and line are incident. This particular graph is a connected cubic graph (regular of degree 3), has girth 6 and each part contains 7 vertices.
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Using a naïve array implementation on a 32-bit computer, an adjacency list for an undirected graph requires about bytes of space, where is the number of edges of the graph. Noting that an undirected simple graph can have at most edges, allowing loops, we can let denote the density of the graph. Then, when , that is the adjacency list representation occupies more space than the adjacency matrix representation when . Thus a graph must be sparse enough to justify an adjacency list representation.
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On the other hand, the property of being empty is non-trivial, because the empty graph possesses this property, but non-empty graphs do not. A graph property is said to be monotone if the addition of edges does not destroy the property. Alternately, if a graph possesses a monotone property, then every supergraph of this graph on the same vertex set also possesses it. For instance, the property of being nonplanar is monotone: a supergraph of a nonplanar graph is itself nonplanar.
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The Euler genus of a graph is the minimal integer n such that the graph can be embedded in an orientable surface of (orientable) genus n/2 or in a non-orientable surface of (non-orientable) genus n. A graph is orientably simple if its Euler genus is smaller than its non-orientable genus. The maximum genus of a graph is the maximal integer n such that the graph can be 2-cell embedded in an orientable surface of genus n.
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The tree-depths of the complete graph K4 and the complete bipartite graph K3,3 are both four, while the tree-depth of the path graph P7 is three. The tree-depth of a complete graph equals its number of vertices. For, in this case, the only possible forest F for which every pair of vertices are in an ancestor-descendant relationship is a single path. Similarly, the tree-depth of a complete bipartite graph Kx,y is min(x,y) + 1\.
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Tietze's graph may be formed from the Petersen graph by replacing one of its vertices with a triangle. Like the Tietze graph, the Petersen graph forms the boundary of six mutually touching regions, but on the projective plane rather than on the Möbius strip. If one cuts a hole from this subdivision of the projective plane, surrounding a single vertex, the surrounded vertex is replaced by a triangle of region boundaries around the hole, giving the previously described construction of the Tietze graph.
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In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a set of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common. Fractional graph coloring can be viewed as the linear programming relaxation of traditional graph coloring.
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The linear arboricity of a graph is the minimum number of linear forests (a collection of paths) into which the edges of the graph can be partitioned. The linear arboricity of a graph is closely related to its maximum degree and its slope number. The pseudoarboricity of a graph is the minimum number of pseudoforests into which its edges can be partitioned. Equivalently, it is the maximum ratio of edges to vertices in any subgraph of the graph, rounded up to an integer.
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In graph theory, the Golomb graph is a polyhedral graph with 10 vertices and 18 edges. It is named after Solomon W. Golomb, who constructed it (with a non- planar embedding) as a unit distance graph that requires four colors in any graph coloring. Thus, like the simpler Moser spindle, it provides a lower bound for the Hadwiger–Nelson problem: coloring the points of the Euclidean plane so that each unit line segment has differently-colored endpoints requires at least four colors.
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The Hoffman graph is not a vertex-transitive graph and its full automorphism group is a group of order 48 isomorphic to the direct product of the symmetric group S4 and the cyclic group Z/2Z. The characteristic polynomial of the Hoffman graph is equal to :(x-4) (x-2)^4 x^6 (x+2)^4 (x+4) making it an integral graph—a graph whose spectrum consists entirely of integers. It is the same spectrum as the hypercube Q4.
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Cambridge [u.a.: Cambridge Univ., 2004. Using vertex splitting, the recognition problem for trapezoid graphs was shown by Mertzios and Corneil to succeed in O(n(n+m)) time, where m denotes the number of edges. This process involves augmenting a given graph {G}, and then transforming the augmented graph by replacing each of the original graph’s vertices by a pair of new vertices. This “split graph” is a permutation graph with special properties if an only if {G} is a trapezoid graph.
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To determine if a graph {G} is a trapezoid graph, search for a transitive orientation {F} on the complement of {G}. Since trapezoid graphs are a subset of co-comparability graphs, if {G} is a trapezoid graph, its complement {G'} must be a comparability graph. If a transitive orientation {F} of the complement {G'} does not exist, {G} is not a trapezoid graph. If {F} does exist, test to see if the order given by {F} is a trapezoid order.
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A well-covered graph, the intersection graph of the nine diagonals of a hexagon. The three red vertices form one of its 14 equal-sized maximal independent sets, and the six blue vertices form the complementary minimal vertex cover. In graph theory, a well-covered graph is an undirected graph in which every minimal vertex cover has the same size as every other minimal vertex cover. Equivalently, these are the graphs in which every maximal independent set has the same size.
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In topological graph theory, the Petrie dual of an embedded graph (on a 2-manifold with all faces disks) is another embedded graph that has the Petrie polygons of the first embedding as its faces. The Petrie dual is also called the Petrial, and the Petrie dual of an embedded graph G may be denoted G^\pi. It can be obtained from a signed rotation system or ribbon graph representation of the embedding by twisting every edge of the embedding.
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A coloring of a given graph is distinguishing for that graph if and only if it is distinguishing for the complement graph. Therefore, every graph has the same distinguishing number as its complement. For every graph , the distinguishing number of is at most proportional to the logarithm of the number of automorphisms of . If the automorphisms form a nontrivial abelian group, the distinguishing number is two, and if it forms a dihedral group then the distinguishing number is at most three.
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A Hamiltonian path (but not cycle) in the Herschel graph As a bipartite graph that has an odd number of vertices, the Herschel graph does not contain a Hamiltonian cycle (a cycle of edges that passes through each vertex exactly once). For, in any bipartite graph, any cycle must alternate between the vertices on either side of the bipartition, and therefore must contain equal numbers of both types of vertex and must have an even length. Thus, a cycle passing once through each of the eleven vertices cannot exist in the Herschel graph. It is the smallest non-Hamiltonian polyhedral graph, whether the size of the graph is measured in terms of its number of vertices, edges, or faces.. There exist other polyhedral graphs with 11 vertices and no Hamiltonian cycles (notably the Goldner–Harary graph.) but none with fewer edges.
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Möbius ladders play an important role in the theory of graph minors. The earliest result of this type is a 1937 theorem of Klaus Wagner (part of a cluster of results known as Wagner's theorem) that graphs with no K5 minor can be formed by using clique-sum operations to combine planar graphs and the Möbius ladder M8. For this reason M8 is called the Wagner graph. The Wagner graph is also one of four minimal forbidden minors for the graphs of treewidth at most three (the other three being the complete graph K5, the graph of the regular octahedron, and the graph of the pentagonal prism) and one of four minimal forbidden minors for the graphs of branchwidth at most three (the other three being K5, the graph of the octahedron, and the cube graph)...
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A hydrogen-depleted molecular graph or hydrogen-suppressed molecular graph is the molecular graph with hydrogen vertices deleted. Molecular graphs can distinguish between structural isomers, compounds which have the same molecular formula but non-isomorphic graphs - such as isopentane and neopentane. On the other hand, the molecular graph normally does not contain any information about the three-dimensional arrangement of the bonds, and therefore cannot distinguish between geometric isomers (such as cis and trans 2-butene) or other stereoisomers (such as D- and L-glyceraldehyde). In some important cases (topological index calculation etc.) the following classical definition is sufficient: molecular graph is connected undirected graph one-to-one corresponded to structural formula of chemical compound so that vertices of the graph correspond to atoms of the molecule and edges of the graph correspond to chemical bonds between these atoms.
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Using this description of the graph, the conjecture may be restated as follows: if some family of sets has total elements, and any two sets intersect in at most one element, then the intersection graph of the sets may be -colored.. The intersection number of a graph is the minimum number of elements in a family of sets whose intersection graph is , or equivalently the minimum number of vertices in a hypergraph whose line graph is . define the linear intersection number of a graph, similarly, to be the minimum number of vertices in a linear hypergraph whose line graph is . As they observe, the Erdős–Faber–Lovász conjecture is equivalent to the statement that the chromatic number of any graph is at most equal to its linear intersection number. present another yet equivalent formulation, in terms of the theory of clones.
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In graph theory, a king's graph is a graph that represents all legal moves of the king chess piece on a chessboard where each vertex represents a square on a chessboard and each edge is a legal move. More specifically, an n \times m king's graph is a king's graph of an n \times m chessboard.. Chang defines the king's graph on p. 341. It is the map graph formed from the squares of a chessboard by making a vertex for each square and an edge for each two squares that share an edge or a corner. It can also be constructed as the strong product of two path graphs.. For an n \times m king's graph the total number of vertices is n m and the number of edges is 4nm -3(n + m) + 2.
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The strong splits of a graph give rise to a structure called the split decomposition or join decomposition of the graph. This decomposition can be represented by a tree whose leaves correspond one- to-one with the given graph, and whose edges correspond one-to-one with the strong splits of the graph, such that the partition of leaves formed by removing any edge from the tree is the same as the partition of vertices given by the associated strong split. Each internal node of the split decomposition tree of a graph is associated with a graph , called the quotient graph for node . The quotient graph can be formed by deleting from the tree, forming subsets of vertices in corresponding to the leaves in each of the resulting subtrees, and collapsing each of these vertex sets into a single vertex.
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The 12-vertex crown graph, the intersection graph of the lines of the double six A generic cubic surface contains 27 lines, among which can be found 36 Schläfli double six configurations. The set of 15 lines complementary to a double six, together with the 15 tangent planes through triples of these lines, has the incidence pattern of another configuration, the Cremona–Richmond configuration. The intersection graph of the twelve lines of the double six configuration is a twelve-vertex crown graph, a bipartite graph in which each vertex is adjacent to five out of the six vertices of the opposite color. The Levi graph of the double six may be obtained by replacing each edge of the crown graph by a two-edge path.
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Alternatively, by the circle packing theorem, any planar graph may be represented as a collection of circles, any two of which cross if and only if the corresponding vertices are adjacent; these circles (with a starting and ending point chosen to turn them into open curves) provide a string graph representation of the given planar graph. proved that every planar graph has a string representation in which each pair of strings has at most one crossing point, unlike the representations described above. Scheinerman's conjecture, now proven, is the even stronger statement that every planar graph may be represented by the intersection graph of straight line segments, a very special case of strings. A subdivision of K5 that is not a string graph.
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For instance in a simple graph, we can define the overlap between two k-cliques to be the number of vertices common to both k-cliques. The Clique Percolation Method is then equivalent to thresholding this clique graph, dropping all edges of weight less than (k-1), with the remaining connected components forming the communities of cliques found in CPM. For k=2 the cliques are the edges of the original graph and the clique graph in this case is the line graph of the original network. In practice, using the number of common vertices as a measure of the strength of clique overlap may give poor results as large cliques in the original graph, those with many more than k vertices, will dominate the clique graph.
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In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particular p. 181.. From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for Friedrich Wilhelm Levi, who wrote about them in 1942.. The Levi graph of a system of points and lines usually has girth at least six: Any 4-cycles would correspond to two lines through the same two points. Conversely any bipartite graph with girth at least six can be viewed as the Levi graph of an abstract incidence structure.
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In graph theory, a branch of mathematics, a crown graph on 2n vertices is an undirected graph with two sets of vertices {u1, u2, ..., un} and {v1, v2, ..., vn} and with an edge from ui to vj whenever i ≠ j. The crown graph can be viewed as a complete bipartite graph from which the edges of a perfect matching have been removed, as the bipartite double cover of a complete graph, as the tensor product Kn × K2, as the complement of the Cartesian direct product of Kn and K2, or as a bipartite Kneser graph Hn,1 representing the 1-item and (n − 1)-item subsets of an n-item set, with an edge between two subsets whenever one is contained in the other.
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The structure of graphs exchanged by GXL streams is given by a schema represented as a Unified Modeling Language (UML) class diagram. Since GXL is a general graph exchange format, it can also be used to interchange any graph-based data, including models between computer-aided software engineering (CASE) tools, data between graph transformation systems, or graph visualization tools. GXL includes support for hypergraphs and hierarchical graphs, and can be extended to support other types of graphs. GXL originated in the merger of GRAph eXchange format (GraX: University of Koblenz, DE) for exchanging typed, attributed, ordered, directed graphs (TGraphs), Tuple Attribute Language (TA: University of Waterloo, CA), and the graph format of the PROGRES graph rewriting system (University Bw München, DE).
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The Möbius–Kantor graph, embedded on the torus. Edges extending upwards from the central square should be viewed as connecting with the corresponding edge extending downwards from the square, and edges extending leftwards from the square should be viewed as connecting with the corresponding edge extending rightwards. The Möbius–Kantor graph cannot be embedded without crossings in the plane; it has crossing number 4, and is the smallest cubic graph with that crossing number . Additionally, it provides an example of a graph all of whose subgraphs' crossing numbers differ from it by two or more.. However, it is a toroidal graph: it has an embedding in the torus in which all faces are hexagons . The dual graph of this embedding is the hyperoctahedral graph K2,2,2,2.
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Golumbic's work in graph theory lead to the study of new perfect graph families such as tolerance graphs, which generalize the classical graph notions of interval graph and comparability graph. He is credited with introducing the systematic study of algorithmic aspects in intersection graph theory, and initiated research on new structured families of graphs including the edge intersection graphs of paths in trees, tolerance graphs, chordal probe graphs and trivially perfect graphs. Golumbic, Kaplan and Shamir introduced the study of graph sandwich problems. In the area of compiler optimization, Golumbic holds a joint patent with Vladimir Rainish, Instruction Scheduler for a Computer, (UK9-90-035/IS), an invention based on their technique called SHACOOF (ScHeduling Across COntrOl Flow), which in Hebrew means "transparent".
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The seven cubic 3-connected well- covered graphs A cubic 1-connected well-covered graph, formed by replacing each node of a six-node path by one of three fragments The snub disphenoid, one of four well-covered 4-connected 3-dimensional simplicial polyhedra. The cubic (3-regular) well-covered graphs have been classified: they consist of seven 3-connected examples, together with three infinite families of cubic graphs with lesser connectivity. The seven 3-connected cubic well-covered graphs are the complete graph , the graphs of the triangular prism and the pentagonal prism, the Dürer graph, the utility graph , an eight-vertex graph obtained from the utility graph by a Y-Δ transform, and the 14-vertex generalized Petersen graph .; ; ; .
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A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible. In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.
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An equivalence class of graphs under switching is called a switching class. Switching was introduced by and developed by Seidel; it has been called graph switching or Seidel switching, partly to distinguish it from switching of signed graphs. In the standard construction of a two-graph from a simple graph given above, two graphs will yield the same two-graph if and only if they are equivalent under switching, that is, they are in the same switching class. Let Γ be a two-graph on the set X. For any element x of X, define a graph with vertex set X having vertices y and z adjacent if and only if {x, y, z} is in Γ. In this graph, x will be an isolated vertex.
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A metric graph is a finite connected graph \gamma together with the assignment to every topological edge e of Γ of a positive real number L(e) > 0 called the length of e. The volume of a metric graph is the sum of the lengths of its topological edges. A marked metric graph structure on Fn consists of a marking f : Rn → Γ together with a metric graph structure L on Γ. Two marked metric graph structures f1 : Rn → Γ1 and f2 : Rn → Γ2 are equivalent if there exists an isometry θ : Γ1 → Γ2 such that, up to free homotopy, we have θ o f1 = f2. The Outer space Xn consists of equivalence classes of all the volume-one marked metric graph structures on Fn.
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The result of a WBA is a why–because graph (WBG). The WBG depicts causal relations between factors of an accident. It is a directed acyclic graph where the nodes of the graph are factors. Directed edges denote cause–effect relations between the factors.
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OGNL uses Java reflection and introspection to address the Object Graph of the runtime application. This allows the program to change behavior based on the state of the object graph instead of relying on compile time settings. It also allows changes to the object graph.
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The friendship graph Fn can be constructed by joining n copies of the cycle graph C3 with a common vertex.Gallian, J. A. "Dynamic Survey DS6: Graph Labeling." Electronic Journal of Combinatorics, DS6, 1-58, January 3, 2007. . The friendship graphs F2, F3 and F4.
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In graph theory, the act of coloring generally implies the assignment of labels to vertices, edges or faces in a graph. The incidence coloring is a special graph labeling where each incidence of an edge with a vertex is assigned a color under certain constraints.
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For 4-regular planar graphs, additional necessary conditions can be derived from Grinberg's theorem. An example of a 4-regular planar graph that does not meet these conditions, and does not have a Hamiltonian decomposition, is given by the medial graph of the Herschel graph.
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In graph theory, the matching polytope of a given graph is a geometric object representing the possible matchings in the graph. It is a convex polytope each of whose corners corresponds to a matching. It has great theoretical importance in the theory of matching.
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According to the Foster census, the Dyck graph, referenced as F32A, is the only cubic symmetric graph on 32 vertices.. The characteristic polynomial of the Dyck graph is equal to (x-3) (x-1)^9 (x+1)^9 (x+3) (x^2-5)^6.
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Knowledge about the structure of the group can be obtained by studying the adjacency matrix of the graph and in particular applying the theorems of spectral graph theory. The genus of a group is the minimum genus for any Cayley graph of that group.
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A graph has only vertices (0-dimensional elements) and edges (1-dimensional elements). We can generalize the graph to an abstract simplicial complex by adding elements of a higher dimension. Then, the concept of graph homology is generalized by the concept of simplicial homology.
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In graph theory, the radius of a graph is the minimum over all vertices u of the maximum distance from u to any other vertex of the graph.Jonathan L. Gross, Jay Yellen (2006), Graph theory and its applications. 2nd edition, 779 pages; CRC Press. , 9781584885054.
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The Petersen Graph is a mathematics book about the Petersen graph and its applications in graph theory. It was written by Derek Holton and John Sheehan, and published in 1993 by the Cambridge University Press as volume 7 in their Australian Mathematical Society Lecture Series.
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A legend at the foot of the graph sheet warns that a point below the latter band indicates "Attention Urgent."Oxford Capacity Analysis personality profile graph. Retrieved 2006 right After the graph has been plotted, a Scientology staff member reviews the results with the testee.
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A graph is called -edge-connected if its edge connectivity is or greater. A graph is said to be maximally connected if its connectivity equals its minimum degree. A graph is said to be maximally edge- connected if its edge-connectivity equals its minimum degree.
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The Gewirtz graph is a strongly regular graph with 56 vertices and valency 10\. It is named after the mathematician Allan Gewirtz, who described the graph in his dissertation.Allan Gewirtz, Graphs with Maximal Even Girth, Ph.D. Dissertation in Mathematics, City University of New York, 1967.
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Perkel graphs with 19-fold symmetry The vertices and edges form the Perkel graph, the unique distance-regular graph with intersection array {6,5,2;1,1,3}, discovered by .
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The distances in the original graph may be calculated from the distances calculated by Dijkstra's algorithm in the reweighted graph by reversing the reweighting transformation.
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Graphs are one of the prime objects of study in discrete mathematics. Refer to the glossary of graph theory for basic definitions in graph theory.
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Behzad, Mehdi, and Gary Chartrand. "No graph is perfect." American Mathematical Monthly (1967): 962–963. Graph theorists thus turned to the issue of local regularity.
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The genus of a group G is the minimum genus of a (connected, undirected) Cayley graph for G. The graph genus problem is NP-complete.
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This is a glossary of graph theory terms. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by edges.
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Typically, graph partition problems fall under the category of NP-hard problems. Solutions to these problems are generally derived using heuristics and approximation algorithms. However, uniform graph partitioning or a balanced graph partition problem can be shown to be NP-complete to approximate within any finite factor. Even for special graph classes such as trees and grids, no reasonable approximation algorithms exist, unless P=NP.
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In mathematics, dependent random choice is a simple yet powerful probabilistic technique which shows how to find a large set of vertices in a dense graph such that every small subset of vertices has a lot of common neighbors. It is a useful tool to embed a graph into another graph with many edges, and thus has its application in extremal graph theory and Ramsey theory.
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Cypher is based on the Property Graph Model, which organizes data into nodes and edges (called “relationships” in Cypher). In addition to those standard graph elements of nodes and relationships, the property graph model adds labels and properties for describing finer categories and attributes of the data. Nodes are the entities in the graph. They can hold any number of attributes (key- value pairs) called properties.
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Graph drawing also can be said to encompass problems that deal with the crossing number and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a planar graph, the crossing number is zero by definition. Drawings on surfaces other than the plane are also studied.
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These averages are plotted onto a specific graph where the intersection of the average number of sentences and the average number of letters/word determines the reading level of the content. Note that this graph is very similar to the Fry readability formula's graph. This graph is primarily used in secondary education to help classify teaching materials and books into their appropriate reading groups.
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Concepts of graph kernels have been around since the 1999, when D. Haussler introduced convolutional kernels on discrete structures. The term graph kernels was more officially coined in 2002 by R. I. Kondor and John Lafferty as kernels on graphs, i.e. similarity functions between the nodes of a single graph, with the World Wide Web hyperlink graph as a suggested application. In 2003, Gaertner et al.
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The incidence between the 35 triplets and 15 Fano planes induces PG(3,2), with 15 points and 35 lines. To make the Hoffman-Singleton graph, create a graph vertex for each of the 15 Fano planes and 35 triplets. Connect each Fano plane to its 7 triplets, like a Levi graph, and also connect disjoint triplets to each other like the odd graph O(4).
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In algebraic graph theory, the adjacency algebra of a graph G is the algebra of polynomials in the adjacency matrix A(G) of the graph. It is an example of a matrix algebra and is the set of the linear combinations of powers of A.Algebraic graph theory, by Norman L. Biggs, 1993, , p. 9 Some other similar mathematical objects are also called "adjacency algebra".
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A similar characterization holds for the closed neighborhood hypergraph: A graph is strongly chordal if and only if the bipartite incidence graph of its closed neighborhood hypergraph is chordal bipartite. Another result found by Elias Dahlhaus is: A bipartite graph B = (X,Y,E) is chordal bipartite if and only if the split graph resulting from making X a clique is strongly chordal., Corollary 8.3.2, p. 129.
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In the mathematical field of graph theory, the ladder graph Ln is a planar undirected graph with 2n vertices and 3n-2 edges. The ladder graph can be obtained as the Cartesian product of two path graphs, one of which has only one edge: Ln,1 = Pn × P2.Hosoya, H. and Harary, F. "On the Matching Properties of Three Fence Graphs." J. Math. Chem.
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A polyhedral graph is the graph of a simple polyhedron if it is cubic (every vertex has three edges), and it is the graph of a simplicial polyhedron if it is a maximal planar graph. The Halin graphs, graphs formed from a planar embedded tree by adding an outer cycle connecting all of the leaves of the tree, form another important subclass of the polyhedral graphs.
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This partial list of graphs contains definitions of graphs and graph families which are known by particular names, but do not have a Wikipedia article of their own. For collected definitions of graph theory terms that do not refer to individual graph types, such as vertex and path, see Glossary of graph theory. For links to existing articles about particular kinds of graphs, see Category:Graphs.
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G4 A gear graph, denoted Gn is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, Gn has 2n+1 vertices and 3n edges. Gear graphs are examples of squaregraphs, and play a key role in the forbidden graph characterization of squaregraphs. Gear graphs are also known as cogwheels and bipartite wheels.
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A cactus graph In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or (for nontrivial cactus) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle.
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In the branch of mathematics called graph theory, the strength of an undirected graph corresponds to the minimum ratio edges removed/components created in a decomposition of the graph in question. It is a method to compute partitions of the set of vertices and detect zones of high concentration of edges, and is analogous to graph toughness which is defined similarly for vertex removal.
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Some classes of planar graphs are necessarily Hamiltonian, and therefore also subhamiltonian; these include the 4-connected planar graphs. and the Halin graphs.. Every planar graph with maximum degree at most four is subhamiltonian,. as is every planar graph with no separating triangles.. If the edges of an arbitrary planar graph are subdivided into paths of length two, the resulting subdivided graph is always subhamiltonian.
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A circular-arc graph (left) and a corresponding arc model (right). In graph theory, a circular-arc graph is the intersection graph of a set of arcs on the circle. It has one vertex for each arc in the set, and an edge between every pair of vertices corresponding to arcs that intersect. Formally, let :I_1, I_2, \ldots, I_n \subset C_1 be a set of arcs.
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Original Drawing The first step is to draw out the precedence constraints in a graphical form called a graph (See Original Drawing picture). Each job originates at the "source", which we will label U on the graph. Each job will finish in a "sink" of jobs, which we will label V on the graph. Each row of nodes in the graph represents a job.
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A 2-vertex-connected graph, its square, and a Hamiltonian cycle in the square In graph theory, a branch of mathematics, Fleischner's theorem gives a sufficient condition for a graph to contain a Hamiltonian cycle. It states that, if G is a 2-vertex-connected graph, then the square of G is Hamiltonian. it is named after Herbert Fleischner, who published its proof in 1974.
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Every Hanoi graph contains a Hamiltonian cycle. The Hanoi graph H^1_k is a complete graph on k vertices. Because they contain complete graphs, all larger Hanoi graphs H^n_k require at least k colors in any graph coloring. They may be colored with exactly k colors by summing the indexes of the towers containing each disk, and using the sum modulo k as the color.
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In graph theory, an induced matching or strong matching is a subset of the edges of an undirected graph that do not share any vertices (it is a matching) and includes every edge connecting any two vertices in the subset (it is an induced subgraph). An induced matching can also be described as an independent set in the square of the line graph of the given graph.
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In graph theory, a bivariegated graph is a graph whose vertex set can be partitioned into two equal parts such that each vertex is adjacent to exactly one vertex from the other set not containing it.... In a bivarigated graph G with 2n vertices, there exists a set of n independent edges such that no odd number of them lie on a cycle of G.
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The smallest hypohamiltonian graph is the Petersen graph . More generally, the generalized Petersen graph GP(n,2) is hypohamiltonian when n is 5 (mod 6); proved that these graphs are non-Hamiltonian, while it is straightforward to verify that their one-vertex deletions are Hamiltonian. See for a classificiation of non-Hamiltonian generalized Petersen graphs. the Petersen graph is the instance of this construction with n = 5\.
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Ellingham, M. N. and Horton, J. D. "Non-Hamiltonian 3-Connected Cubic Bipartite Graphs." J. Combin. Th. Ser. B 34, 350-353, 1983. In 1989, Georges' graph, the smallest currently- known non-Hamiltonian 3-connected cubic bipartite graph was discovered, containing 50 vertices.. As a non-Hamiltonian cubic graph with many long cycles, the Horton graph provides good benchmark for programs that search for Hamiltonian cycles.
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The octahedron, a 3-cross polytope whose edges and vertices form K2,2,2, a Turán graph T(6,3). Unconnected vertices are given the same color in this face-centered projection. Several choices of the parameter r in a Turán graph lead to notable graphs that have been independently studied. The Turán graph T(2n,n) can be formed by removing a perfect matching from a complete graph K2n.
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Therefore, at least four colors are needed to color this graph and the plane containing it. An alternative lower bound in the form of a ten-vertex four-chromatic unit distance graph, the Golomb graph, was discovered at around the same time by Solomon W. Golomb., p. 19. In 2018, computer scientist and biologist Aubrey de Grey found a 1581-vertex, non-4-colourable unit-distance graph.
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It gives a sufficient condition for a graph to be Hamiltonian, essentially stating that a graph with sufficiently many edges must contain a Hamilton cycle. Specifically, the theorem considers the sum of the degrees of pairs of non-adjacent vertices: if every such pair has a sum that at least equals the total number of vertices in the graph, then the graph is Hamiltonian.
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The greedy spanner or greedy graph is defined as the graph resulting from repeatedly adding an edge between the closest pair of points without a t-path. Algorithms which compute this graph are referred to as greedy spanner algorithms. From the construction it trivially follows that the greedy graph is a t-spanner. The greedy spanner was first discovered in 1989 independently by Althöfer and Bern (unpublished).
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The Goldner–Harary graph, an example of a planar 3-tree. In graph theory, a k-tree is an undirected graph formed by starting with a (k + 1)-vertex complete graph and then repeatedly adding vertices in such a way that each added vertex v has exactly k neighbors U such that, together, the k + 1 vertices formed by v and U form a clique.
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A graph with three components. In graph theory, a component, sometimes called a connected component, of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices in the supergraph. For example, the graph shown in the illustration has three components. A vertex with no incident edges is itself a component.
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The matching complex of a complete bipartite graph Km,n is known as a chessboard complex. It is the clique graph of the complement graph of a rook's graph,. and each of its simplices represents a placement of rooks on an m × n chess board such that no two of the rooks attack each other. When m = n ± 1, the chessboard complex forms a pseudo-manifold.
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For example, on a graph with 2 vertices and 1 edge connecting them the pebbling number is 2. No matter how the two pebbles are placed on the vertices of the graph it is always possible to move a pebble to any vertex in the graph. One of the central questions of graph pebbling is the value of π(G) for a given graph G. Other topics in pebbling include cover pebbling, optimal pebbling, domination cover pebbling, bounds, and thresholds for pebbling numbers, deep graphs, and others.
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In graph theory, a branch of mathematics, a linear forest is a kind of forest formed from the disjoint union of path graphs. It is an undirected graph with no cycles in which every vertex has degree at most two. Linear forests are the same thing as claw-free forests. They are the graphs whose Colin de Verdière graph invariant is at most 1.. The linear arboricity of a graph is the minimum number of linear forests into which the graph can be partitioned.
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In the mathematical field of graph theory, the Goldner–Harary graph is a simple undirected graph with 11 vertices and 27 edges. It is named after A. Goldner and Frank Harary, who proved in 1975 that it was the smallest non- Hamiltonian maximal planar graph.. See also the same journal 6(2):33 (1975) and 8:104-106 (1977). Reference from listing of Harary's publications... The same graph had already been given as an example of a non-Hamiltonian simplicial polyhedron by Branko Grünbaum in 1967.
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The vertices of the graph are the right cosets Hg = { hg : h in H } for g in G. The edges of the graph are of the form (Hg,Hgxi). The Cayley graph of the group G with {xi : i in I} is the Schreier coset graph for H = {1G} . A spanning tree of a Schreier coset graph corresponds to a Schreier transversal, as in Schreier's subgroup lemma . The book "Categories and Groupoids" listed below relates this to the theory of covering morphisms of groupoids.
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The Wagner graph is a cubic Hamiltonian graph and can be defined by the LCF notation [4]8. It is an instance of an Andrásfai graph, a type of circulant graph in which the vertices can be arranged in a cycle and each vertex is connected to the other vertices whose positions differ by a number that is 1 (mod 3). It is also isomorphic to the circular clique . It can be drawn as a ladder graph with 4 rungs made cyclic on a topological Möbius strip.
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Several different graph invariants can be defined from the simplest sentences (with different measures of simplicity) that define a given graph. In particular the logical depth of a graph is defined to be the minimum level of nesting of quantifiers (the quantifier rank) in a sentence defining the graph. The sentence outlined above nests the quantifiers for all of its variables, so it has logical depth . The logical width of a graph is the minimum number of variables in a sentence that defines it.
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In the mathematical field of graph theory, the 26-fullerene graph is a polyhedral graph with V = 26 vertices and E = 39 edges. Its planar embedding has three hexagonal faces (including the one shown as the external face of the illustration) and twelve pentagonal faces. As a planar graph with only pentagonal and hexagonal faces, meeting in three faces per vertex, this graph is a fullerene. The existence of this fullerene has been known since at least 1968.. See line 19 of table, p.
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In graph theory, the (a, b)-decomposition of an undirected graph is a partition of its edges into a + 1 sets, each one of them inducing a forest, except one which induces a graph with maximum degree b. If this graph is also a forest, then we call this a F(a, b)-decomposition. A graph with arboricity a is (a, 0)-decomposable. Every (a, 0)-decomposition or (a, 1)-decomposition is a F(a, 0)-decomposition or a F(a, 1)-decomposition respectively.
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Among the generalized Petersen graphs are the n-prism G(n, 1), the Dürer graph G(6, 2), the Möbius-Kantor graph G(8, 3), the dodecahedron G(10, 2), the Desargues graph G(10, 3) and the Nauru graph G(12, 5). Four generalized Petersen graphs – the 3-prism, the 5-prism, the Dürer graph, and G(7, 2) – are among the seven graphs that are cubic, 3-vertex-connected, and well-covered (meaning that all of their maximal independent sets have equal size)..
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The dimension of the Petersen graph is 2. In mathematics, and particularly in graph theory, the dimension of a graph is the least integer such that there exists a "classical representation" of the graph in the Euclidean space of dimension with all the edges having unit length. In a classical representation, the vertices must be distinct points, but the edges may cross one another.Some mathematicians regard this strictly as an "immersion", but many graph theorists, including Erdős, Harary and Tutte, use the term "embedding".
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A collection of unit circles and the corresponding unit disk graph. In geometric graph theory, a unit disk graph is the intersection graph of a family of unit disks in the Euclidean plane. That is, it is a graph with one vertex for each disk in the family, and with an edge between two vertices whenever the corresponding vertices lie within a unit distance of each other. They are commonly formed from a Poisson point process, making them a simple example of a random structure.
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In the mathematical field of graph theory, the McLaughlin graph is a strongly regular graph with parameters (275,112,30,56), and is the only such graph. The group theorist Jack McLaughlin discovered that the automorphism group of this graph had a subgroup of index 2 which was a previously undiscovered finite simple group, now called the McLaughlin sporadic group. The automorphism group has rank 3, meaning that its point stabilizer subgroup divides the remaining 274 vertices into two orbits. Those orbits contain 112 and 162 vertices.
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A cycle k-cover of a graph is a family of cycles which cover every edge of G exactly k times. It has been proven that every bridgeless graph has cycle k-cover for any integer even integer k≥4. For k=2, it is the well-known cycle double cover conjecture is an open problem in graph theory. The cycle double cover conjecture states that in every bridgeless graph there exists a set of cycles that together cover every edge of the graph twice.
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A topological graph is also called a drawing of a graph. An important special class of topological graphs is the class of geometric graphs, where the edges are represented by line segments. (The term geometric graph is sometimes used in a broader, somewhat vague sense.) The theory of topological graphs is an area of graph theory, mainly concerned with combinatorial properties of topological graphs, in particular, with the crossing patterns of their edges. It is closely related to graph drawing, a field which is more application oriented, and topological graph theory, which focuses on embeddings of graphs in surfaces (that is, drawings without crossings).
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If a graph has a cycle double cover, the cycles of the cover can be used to form the 2-cells of a graph embedding onto a two-dimensional cell complex. In the case of a cubic graph, this complex always forms a manifold. The graph is said to be circularly embedded onto the manifold, in that every face of the embedding is a simple cycle in the graph. However, a cycle double cover of a graph with degree greater than three may not correspond to an embedding on a manifold: the cell complex formed by the cycles of the cover may have non-manifold topology at its vertices.
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A directed graph (blue and black) and its condensation (yellow). The strongly connected components (subsets of blue vertices within each yellow vertex) form the blocks of a partition giving rise to the quotient. The condensation of a directed graph is the quotient graph where the strongly connected components form the blocks of the partition. This construction can be used to derive a directed acyclic graph from any directed graph.. The result of one or more edge contractions in an undirected graph G is a quotient of G, in which the blocks are the connected components of the subgraph of G formed by the contracted edges.
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The underlying storage mechanism of graph databases can vary. Some depend on a relational engine and “store” the graph data in a table (although a table is a logical element, therefore this approach imposes another level of abstraction between the graph database, the graph database management system and the physical devices where the data is actually stored). Others use a key- value store or document-oriented database for storage, making them inherently NoSQL structures. Retrieving data from a graph database requires a query language other than SQL, which was designed for the manipulation of data in a relational system and therefore cannot "elegantly" handle traversing a graph.
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Normal spanning trees are also closely related to the ends of an infinite graph, equivalence classes of infinite paths that, intuitively, go to infinity in the same direction. If a graph has a normal spanning tree, this tree must have exactly one infinite path for each of the graph's ends. An infinite graph can be used to form a topological space by viewing the graph itself as a simplicial complex and adding a point at infinity for each end of the graph. With this topology, a graph has a normal spanning tree if and only if its set of vertices can be decomposed into a countable union of closed sets.
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The Herschel graph is a planar graph: it can be drawn in the plane with none of its edges crossing. It is also 3-vertex-connected: the removal of any two of its vertices leaves a connected subgraph. It is a bipartite graph: its vertices can be separated into two subsets of five and six vertices respectively, such that every edge has an endpoint in each subset (the red and blue subsets in the picture). As with any bipartite graph, the Herschel graph is a perfect graph : the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph.
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The Herschel graph is named after British astronomer Alexander Stewart Herschel, who wrote an early paper concerning William Rowan Hamilton's icosian game: the Herschel graph describes the smallest convex polyhedron for which this game has no solution. However, Herschel's paper described solutions for the Icosian game only on the graphs of the regular tetrahedron and regular icosahedron; it did not describe the Herschel graph.. The name "the Herschel graph" makes an early appearance in a graph theory textbook by John Adrian Bondy and U. S. R. Murty, published in 1976. However, the graph itself was described earlier, for instance by H. S. M. Coxeter.
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In graph theory, the thickness of a graph is the minimum number of planar graphs into which the edges of can be partitioned. That is, if there exists a collection of planar graphs, all having the same set of vertices, such that the union of these planar graphs is , then the thickness of is at most ... In other words, the thickness of a graph is the minimum number of planar subgraphs whose union equals to graph .Christian A. Duncan, On Graph Thickness, Geometric Thickness, and Separator Theorems, CCCG 2009, Vancouver, BC, August 17–19, 2009 Thus, a planar graph has thickness 1. Graphs of thickness 2 are called biplanar graphs.
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In mathematics, the Coates graph or Coates flow graph, named after C.L. Coates, is a graph associated with the Coates' method for the solution of a system of linear equations. The Coates graph Gc(A) associated with an n × n matrix A is an n-node, weighted, labeled, directed graph. The nodes, labeled 1 through n, are each associated with the corresponding row/column of A. If entry aji ≠ 0 then there is a directed edge from node i to node j with weight aji. In other words, the Coates graph for matrix A is the one whose adjacency matrix is the transpose of A.
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According to Brooks' theorem every connected cubic graph other than the complete graph K4 can be colored with at most three colors. Therefore, every connected cubic graph other than K4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices. According to Vizing's theorem every cubic graph needs either three or four colors for an edge coloring. A 3-edge-coloring is known as a Tait coloring, and forms a partition of the edges of the graph into three perfect matchings.
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Cubic graphs arise naturally in topology in several ways. For example, if one considers a graph to be a 1-dimensional CW complex, cubic graphs are generic in that most 1-cell attaching maps are disjoint from the 0-skeleton of the graph. Cubic graphs are also formed as the graphs of simple polyhedra in three dimensions, polyhedra such as the regular dodecahedron with the property that three faces meet at every vertex. Representation of a planar embedding as a graph-encoded map An arbitrary graph embedding on a two-dimensional surface may be represented as a cubic graph structure known as a graph-encoded map.
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In graph theory, particularly in the theory of hypergraphs, the line graph of a hypergraph H, denoted L(H), is the graph whose vertex set is the set of the hyperedges of H, with two vertices adjacent in L(H) when their corresponding hyperedges have a nonempty intersection in H. In other words, L(H) is the intersection graph of a family of finite sets. It is a generalization of the line graph of a graph. Questions about line graphs of hypergraphs are often generalizations of questions about line graphs of graphs. For instance, a hypergraph whose edges all have size k is called k-uniform.
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A zero in these locations will be incorrectly interpreted as an edge with no distance, cost, etc. If W is an n \times n matrix containing the edge weights of a graph, then W^k (using this distance product) gives the distances between vertices using paths of length at most k edges, and W^n is the distance matrix of the graph. An arbitrary graph on vertices can be modeled as a weighted complete graph on vertices by assigning a weight of one to each edge of the complete graph that corresponds to an edge of and zero to all other edges. for this complete graph is the adjacency matrix of .
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A maximal planar graph G is a finite simple planar graph to which no more edges can be added while preserving planarity. Such a graph always has a unique planar embedding, in which every face of the embedding (including the outer face) is a triangle. In other words, every maximal planar graph G is the 1-skeleton of a simplicial complex which is homeomorphic to the sphere. The circle packing theorem guarantees the existence of a circle packing with finitely many circles whose intersection graph is isomorphic to G. As the following theorem states more formally, every maximal planar graph can have at most one packing.
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The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most different colors, for a given value of , or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree or .
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A 9-vertex graph in which every edge belongs to a unique triangle and every non-edge is the diagonal of a unique quadrilateral. The 99-graph problem asks for a 99-vertex graph with the same property. In graph theory, Conway's 99-graph problem is an unsolved problem asking whether there exists an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices have exactly two common neighbors. Equivalently, every edge should be part of a unique triangle and every non-adjacent pair should be one of the two diagonals of a unique 4-cycle.
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Among abelian Cayley graphs that are strongly regular and have the last two parameters differing by one, it is the only graph that is not a Paley graph. It is also an integral graph, meaning that the eigenvalues of its adjacency matrix are integers. Like the 9\times 9 Sudoku graph it is an integral abelian Cayley graph whose group elements all have order 3, one of a small number of possibilities for the orders in such a graph. There are five possible combinations of parameters for strongly regular graphs that have one shared neighbor per pair of adjacent vertices and exactly two shared neighbors per pair of non-adjacent vertices.
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A perfect graph is an undirected graph with the property that, in every one of its induced subgraphs, the size of the largest clique equals the minimum number of colors in a coloring of the subgraph. Perfect graphs include many important graphs classes including bipartite graphs, chordal graphs, and comparability graphs. The complement of a graph has an edge between two vertices if and only if the original graph does not have an edge between the same two vertices. Thus, a clique in the original graph becomes an independent set in the complement and a coloring of the original graph becomes a clique cover of the complement.
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By the characterization of interval graphs as AT-free chordal graphs, interval graphs are strongly chordal graphs and hence perfect graphs. Their complements belong to the class of comparability graphs, and the comparability relations are precisely the interval orders. Based on the fact that a graph is an interval graph if and only if it is chordal and its complement is a comparability graph, we have: A graph and its complement are interval graphs if and only if it is both a split graph and a permutation graph. The interval graphs that have an interval representation in which every two intervals are either disjoint or nested are the trivially perfect graphs.
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In the mathematical discipline of graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with n vertices can also be defined as the 1-skeleton of an (n-1)-gonal pyramid. Some authors write Wn to denote a wheel graph with n vertices (n ≥ 4); other authors instead use Wn to denote a wheel graph with n+1 vertices (n ≥ 3), which is formed by connecting a single vertex to all vertices of a cycle of length n. In the rest of this article we use the former notation.
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Clearly, the graph canonization problem is at least as computationally hard as the graph isomorphism problem. In fact, graph isomorphism is even AC0-reducible to graph canonization. However it is still an open question whether the two problems are polynomial time equivalent. While the existence of (deterministic) polynomial algorithms for graph isomorphism is still an open problem in computational complexity theory, in 1977 László Babai reported that with probability at least 1 − exp(−O(n)), a simple vertex classification algorithm produces a canonical labeling of a graph chosen uniformly at random from the set of all n-vertex graphs after only two refinement steps.
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The Theta graph or \Theta-graph belongs to the family of cone- based spanners. The basic method of construction involves partitioning the space around each vertex into a set of cones, which themselves partition the remaining vertices of the graph. Like Yao Graphs, a \Theta-graph contains at most one edge per cone; where they differ is how that edge is selected. Whereas Yao Graphs will select the nearest vertex according to the metric space of the graph, the \Theta-graph defines a fixed ray contained within each cone (conventionally the bisector of the cone) and selects the nearest neighbour with respect to orthogonal projections to that ray.
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An outerplanar graph (or 1-outerplanar graph) has all of its vertices on the unbounded (outside) face of the graph. A 2-outerplanar graph is a planar graph with the property that, when the vertices on the unbounded face are removed, the remaining vertices all lie on the newly formed unbounded face. And so on. More formally, a graph is k-outerplanar if it has a planar embedding such that, for every vertex, there is an alternating sequence of at most k faces and k vertices of the embedding, starting with the unbounded face and ending with the vertex, in which each consecutive face and vertex are incident to each other.
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Meurs Challenger is an online graph visualization program, with data analysis and browsing.. The software supports several graph layout algorithms, and allows the user to interact with the nodes. The displayed data can be filtered using textual search, node and edge type, or based on the graph distance between nodes. Written in ActionScript, the program runs on Windows, Linux, macOS and other platforms that support the Adobe Flash Player. Meurs Challenger was the winner at the 2011 edition of the International Symposium on Graph Drawing, in the large graph category.. It is publicly available as a Facebook application, which displays the network graph of the user's friends.
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The centroid terms are part of this graph, and they thus can be interpreted and scored by inspecting the terms that surround them in the graph.
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Hypre implementation of LOBPCG with multigrid preconditioning has been applied to image segmentation in via spectral graph partitioning using the graph Laplacian for the bilateral filter.
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A star Sk is the complete bipartite graph K1,k. The star S3 is called the claw graph. The star graphs S3, S4, S5 and S6.
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A graph colouring is a subclass of graph labellings. Vertex colourings assign different labels to adjacent vertices, while edge colourings assign different labels to adjacent edges.
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The Graph Giraffe ended in December 2011 featuring a farewell message from Lerner to his readers.Lerner, Yosef. "12/15/2009 - The Graph Giraffe." The Daily Cardinal.
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In the mathematical discipline of graph theory, a rainbow matching in an edge- colored graph is a matching in which all the edges have distinct colors.
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The Gosset graph, named after Thorold Gosset, is a specific regular graph (1-skeleton of the 7-dimensional 321 polytope) with 56 vertices and valency 27..
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Example of Urquhart graph: the (thin cyan) longest edges are removed from each Delaunay triangle. In computational geometry, the Urquhart graph of a set of points in the plane, named after Roderick B. Urquhart, is obtained by removing the longest edge from each triangle in the Delaunay triangulation. The Urquhart graph was described by , who suggested that removing the longest edge from each Delaunay triangle would be a fast way of constructing the relative neighborhood graph (the graph connecting pairs of points p and q when there does not exist any third point r that is closer to both p and q than they are to each other). Since Delaunay triangulations can be constructed in time O(n log n), the same time bound holds for the Urquhart graph as well.. Although it was later shown that the Urquhart graph is not exactly the same as the relative neighborhood graph,.
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The first person to write about the Nauru graph was R. M. Foster, in an effort to collect all the cubic symmetric graphs.. The whole list of cubic symmetric graphs is now named after him the Foster Census and inside this list the Nauru graph is numbered graph F24A but has no specific name.. In 1950, H. S. M. Coxeter cited the graph a second time, giving the Hamiltonian representation used to illustrate this article and describing it as the Levi graph of a projective configuration discovered by Zacharias... In 2003, Ed Pegg wrote in his online MAA column that F24A deserves a name but did not propose one.. Finally, in 2007, David Eppstein used the name Nauru graph because the flag of the Republic of Nauru has a 12-point star similar to the one that appears in the construction of the graph as a generalized Petersen graph.
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Convex and strictly convex grid drawings of the same graph In graph drawing, a convex drawing of a planar graph is a drawing that represents the vertices of the graph as points in the Euclidean plane and the edges as straight line segments, in such a way that all of the faces of the drawing (including the outer face) have a convex boundary. The boundary of a face may pass straight through one of the vertices of the graph without turning; a strictly convex drawing asks in addition that the face boundary turns at each vertex. That is, in a strictly convex drawing, each vertex of the graph is also a vertex of each convex polygon describing the shape of each incident face. Every polyhedral graph has a strictly convex drawing, for instance obtained as the Schlegel diagram of a convex polyhedron representing the graph.
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If G is an undirected graph, then the degeneracy of G is the minimum number p such that every subgraph of G contains a vertex of degree p or smaller. A graph with degeneracy p is called p-degenerate. Equivalently, a p-degenerate graph is a graph that can be reduced to the empty graph by repeatedly removing a vertex of degree p or smaller. It follows from Ramsey's theorem that for any graph G there exists a least integer r(G), the Ramsey number of G, such that any complete graph on at least r(G) vertices whose edges are coloured red or blue contains a monochromatic copy of G. For instance, the Ramsey number of a triangle is 6: no matter how the edges of a complete graph on six vertices are colored red or blue, there is always either a red triangle or a blue triangle.
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A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph. Odd cycle transversal is an NP-complete algorithmic problem that asks, given a graph G = (V,E) and a number k, whether there exists a set of k vertices whose removal from G would cause the resulting graph to be bipartite. The problem is fixed-parameter tractable, meaning that there is an algorithm whose running time can be bounded by a polynomial function of the size of the graph multiplied by a larger function of k.. The name odd cycle transversal comes from the fact that a graph is bipartite if and only if it has no odd cycles. Hence, to delete vertices from a graph in order to obtain a bipartite graph, one needs to "hit all odd cycle", or find a so-called odd cycle transversal set.
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Given a purely combinatorial game, there is an associated rooted directed graph whose vertices are game positions and whose edges are moves, and graph traversal starting from the root is used to create a game tree. If the graph contains directed cycles, then a position in the game could repeat infinitely many times, and rules are usually needed to prevent the game from continuing indefinitely. Otherwise, the graph is a directed acyclic graph, and if it isn't a rooted tree, then the game has transpositions. This graph and its topology is important in the study of game complexity, where the state-space complexity is the number of vertices in the graph, the average game length is the average number of vertices traversed from the root to a vertex with no direct successors, and the average branching factor of a game tree is the average outdegree of the graph.
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Berge conjectured that every Berge graph is perfect, or equivalently that the perfect graphs and the Berge graphs define the same class of graphs. This became known as the strong perfect graph conjecture, until its proof in 2002, when it was renamed the strong perfect graph theorem.
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Graphons are naturally associated with dense simple graphs. There are extensions of this model to dense directed weighted graphs, often referred to as decorated graphons. There are also recent extensions to the sparse graph regime, from both the perspective of random graph models and graph limit theory.
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ONNX provides definitions of an extensible computation graph model, built-in operators and standard data types, focused on inferencing (evaluation). Each computation dataflow graph is a list of nodes that form an acyclic graph. Nodes have inputs and outputs. Each node is a call to an operator.
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Every graph has a cycle basis in which every cycle is an induced cycle. In a 3-vertex-connected graph, there always exists a basis consisting of peripheral cycles, cycles whose removal does not separate the remaining graph., pp. 32, 65.. See in particular Theorem 2.5.
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The Coxeter graph has chromatic number 3, chromatic index 3, radius 4, diameter 4 and girth 7. It is also a 3-vertex- connected graph and a 3-edge-connected graph. It has book thickness 3 and queue number 2.Wolz, Jessica; Engineering Linear Layouts with SAT.
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Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different contexts.
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Graph closed to a range of and fired four torpedoes. Again, explosions convinced the British that hits had been scored and a destroyer probably sunk, but again all the shots had missed. Graph returned to Lerwick on 13 January 1943. Graph undertook no further war patrols.
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XDI (short for "eXtensible Data Interchange") is a semantic data interchange format and protocol under development by the OASIS XDI Technical Committee. The name comes from the addressable graph model XDI uses: every node in the XDI graph is its own RDF graph that is uniquely addressable.
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In the mathematical field of graph theory, the Brinkmann graph is a 4-regular graph with 21 vertices and 42 edges discovered by Gunnar Brinkmann in 1992.Brinkmann, G. "Generating Cubic Graphs Faster Than Isomorphism Checking." Preprint 92-047 SFB 343. Bielefeld, Germany: University of Bielefeld, 1992.
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This relationship is shown on the graphs below. The graph on the left is the cumulative distribution function, which is P(T < t). The graph on the right is P(T > t) = 1 - P(T < t). The graph on the right is the survival function, S(t).
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It has many 9-vertex cliques and requires 9 colors in any graph coloring; a 9-coloring of this graph describes a solved Sudoku puzzle. In contrast, for the Brouwer–Haemers graph, the largest cliques are the triangles and the number of colors needed is 7.
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In the mathematical discipline of graph theory, the edge space and vertex space of an undirected graph are vector spaces defined in terms of the edge and vertex sets, respectively. These vector spaces make it possible to use techniques of linear algebra in studying the graph.
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Alternatively, a graph is outerplanar if and only if it does not contain K4 or K2,3 as a minor, a graph obtained from it by deleting and contracting edges.. A triangle-free graph is outerplanar if and only if it does not contain a subdivision of K2,3.
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3, p. 65. Maximal outerplanar graphs are also formed as the graphs of polygon triangulations. They are examples of 2-trees, of series-parallel graphs, and of chordal graphs. Every outerplanar graph is a circle graph, the intersection graph of a set of chords of a circle.
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In the mathematical field of graph theory, the intersection number of a graph is the smallest number of elements in a representation of as an intersection graph of finite sets. Equivalently, it is the smallest number of cliques needed to cover all of the edges of ...
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Aharoni and Berger generalized Drisko's theorem to any bipartite graph, namely: any family of 2n-1 matchings of size n in a bipartite graph has a rainbow matching of size n. Aharoni, Kotlar and Ziv showed that Drisko's extremal example is unique in any bipartite graph.
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67–69, 1948. A Descartes snark is obtained from the Petersen graph by replacing each vertex with a nonagon and each edge with a particular graph closely related to the Petersen graph. Because there are multiple ways to perform this procedure, there are multiple Descartes snarks.
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The Gosset 321 polytope is a semiregular polytope. Therefore, the automorphism group of the Gosset graph, E7, acts transitively upon its vertices, making it a vertex-transitive graph. The characteristic polynomial of the Gosset graph is. : (x-27)(x-9)^7(x+1)^{27}(x+3)^{21}.
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In graph theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs,, p. 105; , p. 94. and divisor graphs.
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Many other popular random graph models can be understood as exchangeable random graph models defined by some graphon, a detailed survey is included in Orbanz and Roy.
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Ear decompositions may be used to characterize several important graph classes, and as part of efficient graph algorithms. They may also be generalized from graphs to matroids.
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The wheel graph Wn is a graph on n vertices constructed by connecting a single vertex to every vertex in an (n − 1)-cycle. Wheels W_4 – W_9.
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Trapezoids can be used to represent a trapezoid graph by using the definition of trapezoid graph. A trapezoid graph's trapezoid representation can be seen in Figure 1.
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Given a triangle-free planar graph, a 3-coloring of the graph can be found in linear time.. For earlier work on algorithms for this problem, see .
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The Paley graph of order 13, a circulant graph formed as the Cayley graph of Z/13 with generator set {1,3,4} A Cayley graph is a graph defined from a pair (G,S) where G is a group and S is a set of generators for the group; it has a vertex for each group element, and an edge for each product of an element with a generator. In the case of a finite cyclic group, with its single generator, the Cayley graph is a cycle graph, and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite path graph. However, Cayley graphs can be defined from other sets of generators as well. The Cayley graphs of cyclic groups with arbitrary generator sets are called circulant graphs.. These graphs may be represented geometrically as a set of equally spaced points on a circle or on a line, with each point connected to neighbors with the same set of distances as each other point.
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Both edge and vertex contraction techniques are valuable in proof by induction on the number of vertices or edges in a graph, where it can be assumed that a property holds for all smaller graphs and this can be used to prove the property for the larger graph. Edge contraction is used in the recursive formula for the number of spanning trees of an arbitrary connected graph, and in the recurrence formula for the chromatic polynomial of a simple graph. Contractions are also useful in structures where we wish to simplify a graph by identifying vertices that represent essentially equivalent entities. One of the most common examples is the reduction of a general directed graph to an acyclic directed graph by contracting all of the vertices in each strongly connected component.
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Master Thesis, University of Tübingen, 2018 K16 3-coloured as three Clebsch graphs. The edges of the complete graph K16 may be partitioned into three disjoint copies of the 5-regular Clebsch graph. Because the Clebsch graph is a triangle-free graph, this shows that there is a triangle-free three-coloring of the edges of K16; that is, that the Ramsey number R(3,3,3) describing the minimum number of vertices in a complete graph without a triangle-free three-coloring is at least 17. used this construction as part of their proof that R(3,3,3) = 17.. The 5-regular Clebsch graph may be colored with four colors, but not three: its largest independent set has five vertices, not enough to partition the graph into three independent color classes.
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Again, a double cover of the resulting graph may be extended in a straightforward way to a double cover of the original graph: every cycle of the split off graph corresponds either to a cycle of the original graph, or to a pair of cycles meeting at v. Thus, every minimal counterexample must be cubic. But if a cubic graph can have its edges 3-colored (say with the colors red, blue, and green), then the subgraph of red and blue edges, the subgraph of blue and green edges, and the subgraph of red and green edges each form a collection of disjoint cycles that together cover all edges of the graph twice. Therefore, every minimal counterexample must be a non-3-edge-colorable bridgeless cubic graph, that is, a snark.
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The "pearls" of the title include theorems, proofs, problems, and examples in graph theory. It has ten chapters; after an introductory chapter on basic definitions, the remaining chapters material on graph coloring; Hamiltonian cycles and Euler tours; extremal graph theory; subgraph counting problems including connections to permutations, derangements, and Cayley's formula; graph labelings; planar graphs, the four color theorem, and the circle packing theorem; near-planar graphs; and graph embeddings on topological surfaces. The book also includes several unsolved problems such as the Oberwolfach problem on covering complete graphs by cycles, the characterization of magic graphs, and ringel's "earth-moon" problem on coloring biplanar graphs. Despite its subtitle promising "a comprehensive introduction" to graph theory, many important topics in graph theory are not covered, with the selection of topics reflecting author Ringel's research interests.
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Every apex graph has chromatic number at most five: the underlying planar graph requires at most four colors by the four color theorem, and the remaining vertex needs at most one additional color. used this fact in their proof of the case k = 6 of the Hadwiger conjecture, the statement that every 6-chromatic graph has the complete graph K6 as a minor: they showed that any minimal counterexample to the conjecture would have to be an apex graph, but since there are no 6-chromatic apex graphs such a counterexample cannot exist. conjectured that every 6-vertex-connected graph that does not have as a minor must be an apex graph. If this were proved, the Robertson–Seymour–Thomas result on the Hadwiger conjecture would be an immediate consequence.
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The independence complex of a graph is a mathematical object describing the independent sets of the graph. Formally, the independence complex of an undirected graph G, denoted by I(G), is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the independent sets of G. Any subset of an independent set is itself an independent set, so I(G) indeed meets the requirement of an abstract simplicial complex that every subset of a set in the family should also be in the family. Every independent set in a graph is a clique in its complement graph, and vice versa. Therefore, the independence complex of a graph equals the clique complex of its complement graph, and vice versa.
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If G is a graph that contains a subgraph H that is a subdivision of K5 or K3,3, then H is known as a Kuratowski subgraph of G.. With this notation, Kuratowski's theorem can be expressed succinctly: a graph is planar if and only if it does not have a Kuratowski subgraph. The two graphs K5 and K3,3 are nonplanar, as may be shown either by a case analysis or an argument involving Euler's formula. Additionally, subdividing a graph cannot turn a nonplanar graph into a planar graph: if a subdivision of a graph G has a planar drawing, the paths of the subdivision form curves that may be used to represent the edges of G itself. Therefore, a graph that contains a Kuratowski subgraph cannot be planar.
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The Hadwiger conjecture in graph theory proposes that if a graph G does not contain a minor isomorphic to the complete graph on k vertices, then G has a proper coloring with k − 1 colors.. The case k = 5 is a restatement of the four color theorem. The Hadwiger conjecture has been proven for k ≤ 6,. but is unknown in the general case. call it “one of the deepest unsolved problems in graph theory.” Another result relating the four- color theorem to graph minors is the snark theorem announced by Robertson, Sanders, Seymour, and Thomas, a strengthening of the four-color theorem conjectured by W. T. Tutte and stating that any bridgeless 3-regular graph that requires four colors in an edge coloring must have the Petersen graph as a minor.
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A graph G and the corresponding simplex graph κ(G). The blue-colored node in κ(G) corresponds to the zero-vertex clique in G (the empty set), and the magenta node corresponds to the 3-vertex clique. In graph theory, a branch of mathematics, the simplex graph κ(G) of an undirected graph G is itself a graph, with one node for each clique (a set of mutually adjacent vertices) in G. Two nodes of κ(G) are linked by an edge whenever the corresponding two cliques differ in the presence or absence of a single vertex. The empty set is included as one of the cliques of G that are used to form the clique graph, as is every set of one vertex and every set of two adjacent vertices.
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Many examples of problems with checkable algorithms come from graph theory. For instance, a classical algorithm for testing whether a graph is bipartite would simply output a Boolean value: true if the graph is bipartite, false otherwise. In contrast, a certifying algorithm might output a 2-coloring of the graph in the case that it is bipartite, or a cycle of odd length if it is not. Any graph is bipartite if and only if it can be 2-colored, and non- bipartite if and only if it contains an odd cycle.
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Versions of these definitions are also used for directed graphs. Like an undirected graph, a directed graph consists of vertices and edges, but each edge is directed from one of its endpoints to the other endpoint. A directed pseudoforest is a directed graph in which each vertex has at most one outgoing edge; that is, it has outdegree at most one. A directed 1-forest - most commonly called a functional graph (see below), sometimes maximal directed pseudoforest - is a directed graph in which each vertex has outdegree exactly one.
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The Petersen graph is the smallest snark. The flower snark J5 is one of six snarks on 20 vertices. In the mathematical field of graph theory, a snark is a simple, connected, bridgeless cubic graph with chromatic index equal to 4. In other words, it is a graph in which every vertex has three neighbors, the connectivity is redundant so that removing no one edge would split the graph, and the edges cannot be colored by only three colors without two edges of the same color meeting at a point.
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An ear decomposition is odd if each of its ears uses an odd number of edges. A factor-critical graph is a graph with an odd number of vertices, such that for each vertex v, if v is removed from the graph then the remaining vertices have a perfect matching. found that: :A graph G is factor-critical if and only if G has an odd ear decomposition. More generally, a result of makes it possible to find in any graph G the ear decomposition with the fewest even ears.
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Matching diagram for the permutation (4,3,5,1,2), below its corresponding permutation graph In mathematics, a permutation graph is a graph whose vertices represent the elements of a permutation, and whose edges represent pairs of elements that are reversed by the permutation. Permutation graphs may also be defined geometrically, as the intersection graphs of line segments whose endpoints lie on two parallel lines. Different permutations may give rise to the same permutation graph; a given graph has a unique representation (up to permutation symmetry) if it is prime with respect to the modular decomposition., p.191.
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Every Paley graph is self-complementary. For example, the 3 × 3 rook's graph (the Paley graph of order nine) is self-complementary, by a symmetry that keeps the center vertex in place but exchanges the roles of the four side midpoints and four corners of the grid.. All strongly regular self-complementary graphs with fewer than 37 vertices are Paley graphs; however, there are strongly regular graphs on 37, 41, and 49 vertices that are not Paley graphs.. The Rado graph is an infinite self-complementary graph.. See in particular Proposition 5.
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In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph G = (V, E), a perfect matching in G is a subset M of E, such that every vertex in V is adjacent to exactly one edge in M. A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used. Every perfect matching is a maximum-cardinality matching, but the opposite is not true.
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The property graph data model therefore deliberately prevents nesting of graphs, or treating nodes in one graph as edges in another. Each property graph may have a set of labels and a set of properties that are associated with the graph as a whole. Current graph database products and projects often support a limited version of the model described here. For example, Apache Tinkerpop forces each node and each edge to have a single label; Cypher allows nodes to have zero to many labels, but relationships only have a single label (called a reltype).
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Such approaches are often implemented in social network analysis software such as UCInet. The alternative approach is to use cliques of fixed size k. The overlap of these can be used to define a type of k-regular hypergraph or a structure which is a generalisation of the line graph (the case when k=2) known as a "Clique graph". The clique graphs have vertices which represent the cliques in the original graph while the edges of the clique graph record the overlap of the clique in the original graph.
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A Halin graph. In graph theory, a Halin graph is a type of planar graph, constructed by connecting the leaves of a tree into a cycle. The tree must have at least four vertices, none of which has exactly two neighbors; it should be drawn in the plane so none of its edges cross (this is called planar embedding), and the cycle connects the leaves in their clockwise ordering in this embedding. Thus, the cycle forms the outer face of the Halin graph, with the tree inside it.
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In graph theory, a branch of mathematics, a chordal completion of a given undirected graph is a chordal graph, on the same vertex set, that has as a subgraph. A minimal chordal completion is a chordal completion such that any graph formed by removing an edge would no longer be a chordal completion. A minimum chordal completion is a chordal completion with as few edges as possible. A different type of chordal completion, one that minimizes the size of the maximum clique in the resulting chordal graph, can be used to define the treewidth of .
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The weak dual of a plane graph is the subgraph of the dual graph whose vertices correspond to the bounded faces of the primal graph. A plane graph is outerplanar if and only if its weak dual is a forest. For any plane graph , let be the plane multigraph formed by adding a single new vertex in the unbounded face of , and connecting to each vertex of the outer face (multiple times, if a vertex appears multiple times on the boundary of the outer face); then, is the weak dual of the (plane) dual of ..
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The Schlegel diagram of a convex polyhedron represents its vertices and edges as points and line segments in the Euclidean plane, forming a subdivision of an outer convex polygon into smaller convex polygons (a convex drawing of the graph of the polyhedron). It has no crossings, so every polyhedral graph is also a planar graph. Additionally, by Balinski's theorem, it is a 3-vertex-connected graph. According to Steinitz's theorem, these two graph-theoretic properties are enough to completely characterize the polyhedral graphs: they are exactly the 3-vertex-connected planar graphs.
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Since chromatic number is an upper bound on the order of the maximum clique, the latter invariant is also at most degeneracy plus one. By using a greedy coloring algorithm on an ordering with optimal coloring number, one can graph color a k-degenerate graph using at most k + 1 colors.; . A k-vertex-connected graph is a graph that cannot be partitioned into more than one component by the removal of fewer than k vertices, or equivalently a graph in which each pair of vertices can be connected by k vertex-disjoint paths.
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By construction, the collaboration graph is a simple graph, since it has no loop-edges and no multiple edges. The collaboration graph need not be connected. Thus each person who never co- authored a joint paper represents an isolated vertex in the collaboration graph of mathematicians. Both the collaboration graph of mathematicians and movie actors were shown to have "small world topology": they have a very large number of vertices, most of small degree, that are highly clustered, and a "giant" connected component with small average distances between vertices.
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As a Möbius ladder, the Wagner graph is nonplanar but has crossing number one, making it an apex graph. It can be embedded without crossings on a torus or projective plane, so it is also a toroidal graph. It has girth 4, diameter 2, radius 2, chromatic number 3, chromatic index 3 and is both 3-vertex-connected and 3-edge-connected. The Wagner graph has 392 spanning trees; it and the complete graph K3,3 have the most spanning trees among all cubic graphs with the same number of vertices.
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The graph is the benchmark Biegert and Söding used to evaluate homology detection. The benchmark compares CS-BLAST to BLAST using true positives from the same superfamily versus false positive of pairs from different folds [4]. (A GRAPH NEEDS TO GO HERE) The other graph uses detects true positives (with a different scale than the previous graph) and false positives of PSI-BLAST and CSI-BLAST and compares the two for one to five iterations [4]. (A DIFFERENT GRAPH NEEDS TO GO HERE) CS-BLAST offers improved sensitivity and alignment quality in sequence comparison.
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The Nauru graph is Hamiltonian and can be described by the LCF notation : [5, −9, 7, −7, 9, −5]4.Eppstein, D., The many faces of the Nauru graph, 2007. The Nauru graph can also be constructed as the generalized Petersen graph G(12, 5) which is formed by the vertices of a dodecagon connected to the vertices of a twelve-point star in which each point of the star is connected to the points five steps away from it. There is also a combinatorial construction of the Nauru graph.
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The Petersen graph is nonplanar. Any nonplanar graph has as minors either the complete graph K_5, or the complete bipartite graph K_{3,3}, but the Petersen graph has both as minors. The K_5 minor can be formed by contracting the edges of a perfect matching, for instance the five short edges in the first picture. The K_{3,3} minor can be formed by deleting one vertex (for instance the central vertex of the 3-symmetric drawing) and contracting an edge incident to each neighbor of the deleted vertex.
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That is, a proper edge coloring is the same thing as a partition of the graph into disjoint matchings. If the size of a maximum matching in a given graph is small, then many matchings will be needed in order to cover all of the edges of the graph. Expressed more formally, this reasoning implies that if a graph has edges in total, and if at most edges may belong to a maximum matching, then every edge coloring of the graph must use at least different colors., p. 134.
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As with the arboricity, the pseudoarboricity has a matroid structure allowing it to be computed efficiently . The thickness of a graph is the minimum number of planar subgraphs into which its edges can be partitioned. As any planar graph has arboricity three, the thickness of any graph is at least equal to a third of the arboricity, and at most equal to the arboricity. The degeneracy of a graph is the maximum, over all induced subgraphs of the graph, of the minimum degree of a vertex in the subgraph.
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An odd cycle of length greater than cannot be perfect, because its chromatic number is three and its clique number is two. Similarly, the complement of an odd cycle of length cannot be perfect, because its chromatic number is and its clique number is . (Alternatively, the imperfection of this graph follows from the perfect graph theorem and the imperfection of the complementary odd cycle). Because these graphs are not perfect, every perfect graph must be a Berge graph, a graph with no odd holes and no odd antiholes.
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A directed cycle graph of length 8 A directed cycle graph is a directed version of a cycle graph, with all the edges being oriented in the same direction. In a directed graph, a set of edges which contains at least one edge (or arc) from each directed cycle is called a feedback arc set. Similarly, a set of vertices containing at least one vertex from each directed cycle is called a feedback vertex set. A directed cycle graph has uniform in-degree 1 and uniform out-degree 1\.
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The Laves graph is a cubic graph (there are exactly three edges at each vertex) and a symmetric graph (every incident pair of a vertex and an edge can be transformed into every other such pair by a symmetry of the graph). The girth of this structure is 10 — the shortest cycles in the graph have 10 vertices — and 15 of these cycles pass through each vertex. The cells of the Voronoi diagram of this structure are heptadecahedra with 17 faces each. They are plesiohedra, polyhedra that tile space isohedrally.
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The complete graph K5 has a RAC drawing with straight edges, but K6 does not. Every 6-vertex RAC drawing has at most 14 edges, but K6 has 15 edges, too many to have a RAC drawing. A complete bipartite graph Ka,b has a RAC drawing with straight edges if and only if either min(a,b) ≤ 2 or a + b ≤ 7\. If min(a,b) ≤ 2, then the graph is a planar graph, and (by Fáry's theorem) every planar graph has a straight-line drawing with no crossings.
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Ore's theorem is a generalization of Dirac's theorem that, when each vertex has degree at least , the graph is Hamiltonian. For, if a graph meets Dirac's condition, then clearly each pair of vertices has degrees adding to at least . In turn Ore's theorem is generalized by the Bondy–Chvátal theorem. One may define a closure operation on a graph in which, whenever two nonadjacent vertices have degrees adding to at least , one adds an edge connecting them; if a graph meets the conditions of Ore's theorem, its closure is a complete graph.
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A graph is defined to be k-ultrahomogeneous if every isomorphism between two of its induced subgraphs of at most k vertices can be extended to an automorphism of the whole graph. If a graph is 5-ultrahomogeneous, it is ultrahomogeneous for every k; the only finite connected graphs of this type are complete graphs, Turán graphs, 3 × 3 rook's graphs, and the 5-cycle. The infinite Rado graph is countably ultrahomogeneous. There are only two connected graphs that are 4-ultrahomogeneous but not 5-ultrahomogeneous: the Schläfli graph and its complement.
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Three friendship graphs, examples of non-Hamiltonian factor- critical graphs The gyroelongated pentagonal pyramid, a claw-free factor- critical graph Any odd-length cycle graph is factor-critical, as is any complete graph with an odd number of vertices. More generally, every Hamiltonian graph with an odd number of vertices is factor-critical. The friendship graphs (graphs formed by connecting a collection of triangles at a single common vertex) provide examples of graphs that are factor-critical but not Hamiltonian. If a graph is factor-critical, then so is the Mycielskian of .
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Mycielskian construction applied to a 5-cycle graph, producing the Grötzsch graph with 11 vertices and 20 edges, the smallest triangle-free 4-chromatic graph . Let the n vertices of the given graph G be v1, v2, . . . , vn. The Mycielski graph μ(G) contains G itself as a subgraph, together with n+1 additional vertices: a vertex ui corresponding to each vertex vi of G, and an extra vertex w. Each vertex ui is connected by an edge to w, so that these vertices form a subgraph in the form of a star K1,n.
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The powers of three give the place values in the ternary numeral system. In graph theory, powers of three appear in the Moon–Moser bound on the number of maximal independent sets of an -vertex graph, and in the time analysis of the Bron–Kerbosch algorithm for finding these sets. Several important strongly regular graphs also have a number of vertices that is a power of three, including the Brouwer–Haemers graph (81 vertices), Berlekamp–van Lint–Seidel graph (243 vertices), and Games graph (729 vertices).For the Brouwer–Haemers and Games graphs, see .
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In graph theory, an arborescence is a directed graph in which, for a vertex u called the root and any other vertex v, there is exactly one directed path from u to v. An arborescence is thus the directed-graph form of a rooted tree, understood here as an undirected graph. Equivalently, an arborescence is a directed, rooted tree in which all edges point away from the root; a number of other equivalent characterizations exist. Every arborescence is a directed acyclic graph (DAG), but not every DAG is an arborescence.
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Chordless cycles may be used to characterize perfect graphs: by the strong perfect graph theorem, a graph is perfect if and only if none of its holes or antiholes have an odd number of vertices that is greater than three. A chordal graph, a special type of perfect graph, has no holes of any size greater than three. The girth of a graph is the length of its shortest cycle; this cycle is necessarily chordless. Cages are defined as the smallest regular graphs with given combinations of degree and girth.
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A vertex cut or separating set of a connected graph is a set of vertices whose removal renders disconnected. The vertex connectivity (where is not a complete graph) is the size of a minimal vertex cut. A graph is called ''-vertex-connected or ''-connected if its vertex connectivity is or greater. More precisely, any graph (complete or not) is said to be -vertex-connected if it contains at least vertices, but does not contain a set of vertices whose removal disconnects the graph; and is defined as the largest such that is -connected.
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For the general truncus form above, the constant a dilates the graph by a factor of a from the x-axis; that is, the graph is stretched vertically when a > 1 and compressed vertically when 0 < a < 1\. When a < 0 the graph is reflected in the x-axis as well as being stretched vertically. The constant b translates the graph horizontally left b units when b > 0, or right when b < 0\. The constant c translates the graph vertically up c units when c > 0 or down when c < 0\.
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A graph with a loop having vertices labeled by degree In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice.Diestel p.5 The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G, denoted by \Delta(G), and the minimum degree of a graph, denoted by \delta(G), are the maximum and minimum degree of its vertices.
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The exception of Whitney's theorem: these two graphs are not isomorphic but have isomorphic line graphs. The Whitney graph isomorphism theorem, shown by Hassler Whitney, states that two connected graphs are isomorphic if and only if their line graphs are isomorphic, with a single exception: K3, the complete graph on three vertices, and the complete bipartite graph K1,3, which are not isomorphic but both have K3 as their line graph. The Whitney graph theorem can be extended to hypergraphs.Dirk L. Vertigan, Geoffrey P. Whittle: A 2-Isomorphism Theorem for Hypergraphs.
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Fortunately, there exists such a recursive decomposition of a graph that implicitly represents all ways of decomposing it; this is the modular decomposition. It is itself a way of decomposing a graph recursively into quotients, but it subsumes all others. The decomposition depicted in the figure below is this special decomposition for the given graph. A graph, its quotient where "bags" of vertices of the graph correspond to the children of the root of the modular decomposition tree, and its full modular decomposition tree: series nodes are labeled "s", parallel nodes "//" and prime nodes "p".
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Constraint satisfaction problems composed of binary constraints only can be viewed as graphs, where the vertices are variables and the edges represent the presence of a constraint between two variables. This graph is called the Gaifman graph or primal constraint graph (or simply primal graph) of the problem. If the primal graph of a problem is acyclic, establishing satisfiability of the problem is a tractable problem. This is a structural restriction, as it can be checked by looking only at the scopes of the constraints, disregarding their relations and the domain.
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Melencolia I by Albrecht Dürer, the first appearance of Dürer's solid (1514). In the mathematical field of graph theory, the Dürer graph is an undirected graph with 12 vertices and 18 edges. It is named after Albrecht Dürer, whose 1514 engraving Melencolia I includes a depiction of Dürer's solid, a convex polyhedron having the Dürer graph as its skeleton. Dürer's solid is one of only four well-covered simple convex polyhedra.
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On the right is shown the reweighted graph, formed by replacing each edge weight by . In this reweighted graph, all edge weights are non-negative, but the shortest path between any two nodes uses the same sequence of edges as the shortest path between the same two nodes in the original graph. The algorithm concludes by applying Dijkstra's algorithm to each of the four starting nodes in the reweighted graph.
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Finally, a graph is Hamiltonian if there exists a cycle that passes through each of its vertices exactly once. Barnette's conjecture states that every cubic bipartite polyhedral graph is Hamiltonian. By Steinitz's theorem, a planar graph represents the edges and vertices of a convex polyhedron if and only if it is polyhedral. A three-dimensional polyhedron has a cubic graph if and only if it is a simple polyhedron.
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A bigraph (often used in the plural bigraphs) can be modelled as the superposition of a graph (the link graph) and a set of trees (the place graph).A Brief Introduction To Bigraphs, IT University of Copenhagen, Denmark.Milner, Robin. The Bigraphical Model, University of Cambridge Computer Laboratory, UK. Each node of the bigraph is part of a graph and also part of some tree that describes how the nodes are nested.
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A general approach when calculating multiple sequence alignments is to use graphs to identify all of the different alignments. When finding alignments via graph, a complete alignment is created in a weighted graph that contains a set of vertices and a set of edges. Each of the graph edges has a weight based on a certain heuristic that helps to score each alignment or subset of the original graph.
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The Journal of Graph Theory is a peer-reviewed mathematics journal specializing in graph theory and related areas, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. It is published by John Wiley & Sons. The journal was established in 1977 by Frank Harary.
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They reduce the graph coloring problem to the register allocation problem by showing that for an arbitrary graph, a program can be constructed such that the register allocation for the program (with registers representing nodes and machine registers representing available colors) would be a coloring for the original graph. As Graph Coloring is an NP-Hard problem and Register Allocation is in NP, this proves the NP-completeness of the problem.
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Elementary operations or editing operations, which are also known as graph edit operations, create a new graph from one initial one by a simple local change, such as addition or deletion of a vertex or of an edge, merging and splitting of vertices, edge contraction, etc. The graph edit distance between a pair of graphs is the minimum number of elementary operations required to transform one graph into the other.
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The automorphism group of the Higman–Sims graph is a group of order isomorphic to the semidirect product of the Higman–Sims group of order with the cyclic group of order 2. It has automorphisms that take any edge to any other edge, making the Higman–Sims graph an edge-transitive graph.Brouwer, A. E. and Haemers, W. H. "The Gewirtz Graph: An Exercise in the Theory of Graph Spectra." Euro.
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An ergograph is a graph that shows a relation between human activities and a seasonal year. The name was coined by Dr. Arthur Geddes of the University of Edinburgh. It can either be a polar coordinate (circular) or a cartesian coordinate (rectangular) graph, and either a line graph or a bar graph. In polar form, the months of the year are marked around the circumference, forming 30° sectors.
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In a random graph model of spacetime, points in space or events in spacetime are represented by nodes of a graph. Each node may be connected to any other node by a link. In mathematical terms this structure is called a graph. The smallest number of links that it takes to go between two nodes of the graph can be interpreted as a measure of the distance between them in space.
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The Biggs-Smith graph is also uniquely determined by its graph spectrum, the set of graph eigenvalues of its adjacency matrix.E. R. van Dam and W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs. J. Algebraic Combin. 15, pages 189-202, 2003 The characteristic polynomial of the Biggs-Smith graph is : (x-3) (x-2)^{18} x^{17} (x^2-x-4)^9 (x^3+3 x^2-3)^{16}.
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In the mathematical field of graph theory, the Dyck graph is a 3-regular graph with 32 vertices and 48 edges, named after Walther von Dyck.. It is Hamiltonian with 120 distinct Hamiltonian cycles. It has chromatic number 2, chromatic index 3, radius 5, diameter 5 and girth 6. It is also a 3-vertex- connected and a 3-edge-connected graph. It has book thickness 3 and queue number 2.
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The shown graph appears as a subgraph of a graph G if, and only if. G satisfies the formula \exists x_1, x_2, x_3, x_4. x_1 \sim x_2 \land x_2 \sim x_3 \land x_3 \sim x_1 \land x_3 \sim x_4. In the first-order logic of graphs, a graph property is expressed as a quantified logical formula whose variables represent graph vertices, with predicates for equality and adjacency testing.
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Such a graph is sometimes also called 1-arc-transitive or flag-transitive. By definition (ignoring u1 and u2), a symmetric graph without isolated vertices must also be vertex-transitive. Since the definition above maps one edge to another, a symmetric graph must also be edge-transitive. However, an edge-transitive graph need not be symmetric, since a—b might map to c—d, but not to d—c.
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For instance, , that is, if the edges of the graph are 2-colored, there will always be a monochromatic triangle. A path in an edge-colored graph is said to be a rainbow path if no color repeats on it. A graph is said to be rainbow colored if there is a rainbow path between any two pairs of vertices. An edge-colouring of a graph G with colours 1. . .
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There are eight ways that signs can be assigned to the sides of a triangle. An odd number of negative signs makes an unbalanced triangle, according to Fritz Heider's theory. In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign. A signed graph is balanced if the product of edge signs around every cycle is positive.
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Miroslav Fiedler (7 April 1926 – 20 November 2015) was a Czech mathematician known for his contributions to linear algebra, graph theory and algebraic graph theory. His article, "Algebraic Connectivity of Graphs", published in the Czechoslovak Math Journal in 1973, established the use of the eigenvalues of the Laplacian matrix of a graph to create tools for measuring algebraic connectivity in algebraic graph theory.Algebraic connectivity of graphs. Czechoslovak Math.
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All such graphs are bipartite, and hence can be colored with only two colors. An edge-transitive graph that is also regular, but not vertex-transitive, is called semi-symmetric. The Gray graph again provides an example. Every edge- transitive graph that is not vertex-transitive must be bipartite and either semi-symmetric or biregular.. The vertex connectivity of an edge-transitive graph always equals its minimum degree.
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In graph drawing, Mutzel has contributed in work on planarization, crossing minimization in layered graph drawing, and SPQR trees, and co-edited a book on graph drawing. She was both the program chair and organizational chair of the 9th International Symposium on Graph Drawing, in Vienna in 2001.GD 2001 web site, retrieved 2014-07-04. Mutzel's other contributions include works on the Ising model, steganography, and Steiner trees.
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In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane, who published it in 1937. It states that a finite undirected graph is planar if and only if the cycle space of the graph (taken modulo 2) has a cycle basis in which each edge of the graph participates in at most two basis vectors.
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In the mathematical discipline of graph theory, a feedback vertex set of a graph is a set of vertices whose removal leaves a graph without cycles. In other words, each feedback vertex set contains at least one vertex of any cycle in the graph. The feedback vertex set problem is an NP-complete problem in computational complexity theory. It was among the first problems shown to be NP-complete.
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An outerplanar graph is biconnected if and only if the outer face of the graph forms a simple cycle without repeated vertices. An outerplanar graph is Hamiltonian if and only if it is biconnected; in this case, the outer face forms the unique Hamiltonian cycle.; . More generally, the size of the longest cycle in an outerplanar graph is the same as the number of vertices in its largest biconnected component.
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The Pappus graph The Levi graph of the Pappus configuration is known as the Pappus graph. It is a bipartite symmetric cubic graph with 18 vertices and 27 edges., p. 28. The Desargues configuration can also be defined in terms of perspective triangles, and the Reye configuration can be defined analogously from two tetrahedra that are in perspective with each other in four different ways, forming a desmic system of tetrahedra.
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Like the usual dual graph, repeating the Petrie dual operation twice returns to the original surface embedding. Unlike the usual dual graph (which is an embedding of a generally different graph in the same surface) the Petrie dual is an embedding of the same graph in a generally different surface. Surface duality and Petrie duality are two of the six Wilson operations, and together generate the group of these operations.
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A de Bruijn graph. With the vertex ordering shown, the partition of the edges into two subsets looping around the left and right sides of the drawing is a 2-queue layout of this graph. In mathematics, the queue number of a graph is a graph invariant defined analogously to stack number (book thickness) using first-in first-out (queue) orderings in place of last-in first-out (stack) orderings.
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In graph theory, a clique cover or partition into cliques of a given undirected graph is a partition of the vertices of the graph into cliques, subsets of vertices within which every two vertices are adjacent. A minimum clique cover is a clique cover that uses as few cliques as possible. The minimum k for which a clique cover exists is called the clique cover number of the given graph.
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The graph of an octahedron is complete multipartite () and well-colored. In graph theory, a subfield of mathematics, a well-colored graph is an undirected graph for which greedy coloring uses the same number of colors regardless of the order in which colors are chosen for its vertices. That is, for these graphs, the chromatic number (minimum number of colors) and Grundy number (maximum number of greedily-chosen colors) are equal.
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The bond can be seen below. 318x318px Next you start the process of simplifying the bond graph, by removing the 1-junction of the wall, and removing junctions with less than three bonds. The bond graph can be seen below. 375x375px There is parallel power in the bond graph.
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A group-interval scheduling problem, i.e. GISMPk, can be described by a similar interval-intersection graph, with additional edges between each two intervals of the same group, i.e., this is the edge union of an interval graph and a graph consisting of n disjoint cliques of size k.
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The main phases in a Chaitin-style graph-coloring register allocator are: Chaitin et al.'s iterative graph coloring based register allocator # Renumber: discover live range information in the source program. # Build: build the interference graph. # Coalesce: merge the live ranges of non-interfering variables related by copy instructions.
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Miquel configuration Rhombic dodecahedral graph In geometry, the Miquel configuration is a configuration of eight points and six circles in the Euclidean plane, with four points per circle and three circles through each point.. Its Levi graph is the Rhombic dodecahedral graph. The configuration is related to Miquel's theorem.
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In this context, the term graph means multigraph. There are several ways to define series-parallel graphs. The following definition basically follows the one used by David Eppstein. A two- terminal graph (TTG) is a graph with two distinguished vertices, s and t called source and sink, respectively.
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In the mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. It is one of the 13 known cubic distance- regular graphs.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular Graphs. New York: Springer-Verlag, 1989.
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Master Thesis, University of Tübingen, 2018 The Coxeter graph is hypohamiltonian: it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It has rectilinear crossing number 11, and is the smallest cubic graph with that crossing number .
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At the 1923 Thermodynamics Conference held in Los Angeles, it was decided to name, in his honor, as a “Mollier graph” any thermodynamic diagram using the Enthalpy h as one of its axes. Example: the h–s graph for steam or the h–x graph for moist air.
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These 27 vectors correspond to the vertices of the Schläfli graph; two vertices are adjacent if and only if the corresponding two vectors have 1 as their inner product.. Alternately, this graph can be seen as the complement of the collinearity graph of the generalized quadrangle GQ(2,4).
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The automorphism group of the Folkman graph acts transitively on its edges but not on its vertices. It is the smallest undirected graph that is edge-transitive and regular, but not vertex- transitive.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp.
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It is NP-complete to determine whether a given graph is a tolerance graph. However, because tolerance graphs are perfect graphs, many algorithmic problems that are hard on other classes of graphs, including graph coloring and the clique problem, can be solved in polynomial time on tolerance graphs.
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In computer science, an attributed graph grammar is a class of graph grammar that associates vertices with a set of attributes and rewrites with functions on attributes. In the algebraic approach to graph grammars, they are usually formulated using the double-pushout approach or the single-pushout approach.
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In the mathematical discipline of graph theory, a graph C is a covering graph of another graph G if there is a covering map from the vertex set of C to the vertex set of G. A covering map f is a surjection and a local isomorphism: the neighbourhood of a vertex v in C is mapped bijectively onto the neighbourhood of f(v) in G. The term lift is often used as a synonym for a covering graph of a connected graph. Though it may be misleading, there is no (obvious) relationship between covering graph and vertex cover or edge cover. The combinatorial formulation of covering graphs is immediately generalized to the case of multigraphs. If we identify a multigraph with a 1-dimensional cell complex, a covering graph is nothing but a special example of covering spaces of topological spaces, so that the terminology in the theory of covering spaces is available; say covering transformation group, universal covering, abelian covering, and maximal abelian covering.
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In the mathematical field of graph theory, the Chvátal graph is an undirected graph with 12 vertices and 24 edges, discovered by . It is triangle-free: its girth (the length of its shortest cycle) is four. It is 4-regular: each vertex has exactly four neighbors. And its chromatic number is 4: it can be colored using four colors, but not using only three. It is, as Chvátal observes, the smallest possible 4-chromatic 4-regular triangle-free graph; the only smaller 4-chromatic triangle-free graph is the Grötzsch graph, which has 11 vertices but has maximum degree 5 and is not regular. This graph is not vertex- transitive: the automorphisms group has one orbit on vertices of size 8, and one of size 4. By Brooks’ theorem, every k-regular graph (except for odd cycles and cliques) has chromatic number at most k. It was also known since that, for every k and l there exist k-chromatic graphs with girth l.
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A planar graph is a graph whose vertices can be represented by points in the Euclidean plane, and whose edges can be represented by simple curves in the same plane connecting the points representing their endpoints, such that no two curves intersect except at a common endpoint. Planar graphs are often drawn with straight line segments representing their edges, but by Fáry's theorem this makes no difference to their graph-theoretic characterization. A subdivision of a graph is a graph formed by subdividing its edges into paths of one or more edges. Kuratowski's theorem states that a finite graph G is planar, if it is not possible to subdivide the edges of K5 or K3,3, and then possibly add additional edges and vertices, to form a graph isomorphic to G. Equivalently, a finite graph is planar if and only if it does not contain a subgraph that is homeomorphic to K5 or K3,3.
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Given a channel, a pair of two horizontal lines, a trapezoid between these lines is defined by two points on the top and two points on the bottom line. A graph is a trapezoid graph if there exists a set of trapezoids corresponding to the vertices of the graph such that two vertices are joined by an edge if and only if the corresponding trapezoids intersect. The interval order dimension of a partially ordered set, P=(X, <), is the minimum number d of interval orders P1 … Pd such that P = P1∩…∩Pd. The incomparability graph of a partially ordered set P=(X, <) is the undirected graph G=(X, E) where x is adjacent to y in G if and only if x and y are incomparable in P. An undirected graph is a trapezoid graph if and only if it is the incomparability graph of a partial order having interval order dimension at most 2.
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The two-graphs on X consisting of all possible triples of X and no triples of X are regular two-graphs and are considered to be trivial two-graphs. For non- trivial two-graphs on the set X, the two-graph is regular if and only if for some x in X the graph Γx is a strongly regular graph with k = 2μ (the degree of any vertex is twice the number of vertices adjacent to both of any non- adjacent pair of vertices). If this condition holds for one x in X, it holds for all the elements of X. It follows that a non-trivial regular two-graph has an even number of points. If G is a regular graph whose two-graph extension is Γ having n points, then Γ is a regular two-graph if and only if G is a strongly regular graph with eigenvalues k, r and s satisfying n = 2(k - r) or n = 2(k - s).
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All algorithmic applications of bidimensionality are based on the following combinatorial property: either the treewidth of a graph is small, or the graph contains a large grid as a minor or contraction. More precisely, # There is a function f such that every graph G excluding a fixed h-vertex graph as a minor and of treewidth at least f(h)r contains (r x r)-grid as a minor. # There is a function g such that every graph G excluding a fixed h-vertex apex graph as a minor and of treewidth at least g(h) r can be edge-contracted to \Gamma_r. Halin's grid theorem is an analogous excluded grid theorem for infinite graphs.
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Graph structure theorem. For any graph H, there exists a positive integer k such that every H-free graph can be obtained as follows: # We start with a list of graphs, where each graph in the list is embedded on a surface on which H does not embed # to each embedded graph in the list, we add at most k vortices, where each vortex has depth at most k # to each resulting graph we add at most k new vertices (called apexes) and add any number of edges, each having at least one of its endpoints among the apexes. # finally, we join via k-clique-sums the resulting list of graphs. Note that steps 1.
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Therefore, the best known time bound for testing whether a graph is triangle- free, O(m1.41),, based on fast matrix multiplication. Here m is the number of edges in the graph, and the big O notation hides a large constant factor; the best practical algorithms for triangle detection take time O(m3/2). For median graph recognition, the time bound can be expressed either in terms of m or n (the number of vertices), as m = O(n log n). applies as well to testing whether a graph is a median graph, and any improvement in median graph testing algorithms would also lead to an improvement in algorithms for detecting triangles in graphs.
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By Steinitz's theorem, the Goldner–Harary graph is a polyhedral graph: it is planar and 3-connected, so there exists a convex polyhedron having the Goldner–Harary graph as its skeleton. Geometrically, a polyhedron representing the Goldner–Harary graph may be formed by gluing a tetrahedron onto each face of a triangular dipyramid, similarly to the way a triakis octahedron is formed by gluing a tetrahedron onto each face of an octahedron. That is, it is the Kleetope of the triangular dipyramid.. Same page, 2nd ed., Graduate Texts in Mathematics 221, Springer-Verlag, 2003, .. The dual graph of the Goldner–Harary graph is represented geometrically by the truncation of the triangular prism.
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More precisely, given a vertex set V, a mandatory edge set E1, and a larger edge set E2, a graph G = (V, E) is called a sandwich graph for the pair G1 = (V, E1), G2 = (V, E2) if E1 ⊆ E ⊆ E2. The graph sandwich problem for property Π is defined as follows:.. :Graph Sandwich Problem for Property Π: :Instance: Vertex set V and edge sets E1 ⊆ E2 ⊆ V × V. :Question: Is there a graph G = (V, E) such that E1 ⊆ E ⊆ E2 and G satisfies property Π ? The recognition problem for a class of graphs (those satisfying a property Π) is equivalent to the particular graph sandwich problem where E1 = E2, that is, the optional edge set is empty.
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Expanding a vertex of a 2k-regular graph into a clique of 2k vertices, one for each endpoint of an edge at the replaced vertex, cannot change whether the graph has a Hamiltonian decomposition. The reverse of this expansion process, collapsing a clique to a single vertex, will transform any Hamiltonian decomposition in the larger graph into a Hamiltonian decomposition in the original graph. Conversely, Walecki's construction can be applied to the clique to expand any Hamiltonian decomposition of the smaller graph into a Hamiltonian decomposition of the expanded graph. This expansion process can be used to produce arbitrarily large vertex-transitive graphs and Cayley graphs of even degree that do not have Hamiltonian decompositions.
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A matching-critical graph is a graph for which the removal of any vertex does not change the size of a maximum matching; by Gallai's characterization, the matching-critical graphs are exactly the graphs in which every connected component is factor-critical.. The complement graph of a critical graph is necessarily matching-critical, a fact that was used by Gallai to prove lower bounds on the number of vertices in a critical graph.. As cited by . Beyond graph theory, the concept of factor-criticality has been extended to matroids by defining a type of ear decomposition on matroids and defining a matroid to be factor-critical if it has an ear decomposition in which all ears are odd..
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Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena.Danail Bonchev, D.H. Rouvray (eds.) (1991) "Chemical Graph Theory: Introduction and Fundamentals", The pioneers of chemical graph theory are Alexandru Balaban, Ante Graovac, Iván Gutman, Haruo Hosoya, Milan Randić and Nenad TrinajstićNenad Trinajstić – Pioneer of Chemical Graph Theory , by Milan Randić (also Harry Wiener and others). In 1988, it was reported that several hundred researchers worked in this area producing about 500 articles annually. A number of monographs have been written in the area, including the two-volume comprehensive text by Trinajstić, Chemical Graph Theory, that summarized the field up to mid-1980s.
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A planar embedding of a given graph is a drawing of the graph in the Euclidean plane, with points for its vertices and curves for its edges, in such a way that the only intersections between pairs of edges are at a common endpoint of the two edges. A minor of a given graph is another graph formed by deleting vertices, deleting edges, and contracting edges. When an edge is contracted, its two endpoints are merged to form a single vertex. In some versions of graph minor theory the graph resulting from a contraction is simplified by removing self-loops and multiple adjacencies, while in other version multigraphs are allowed, but this variation makes no difference to Wagner's theorem.
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The Errera graph is planar and has chromatic number 4, chromatic index 6, radius 3, diameter 4 and girth 3. All its vertices are of degree 5 or 6 and it is a 5-vertex-connected graph and a 5-edge-connected graph. The Errera graph is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 20, the group of symmetries of a decagon, including both rotations and reflections. The characteristic polynomial of the Errera graph is -(x^2-2 x-5) (x^2+x-1)^2 (x^3-4 x^2-9 x+10) (x^4+2 x^3-7 x^2-18 x-9)^2.
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A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles. The line graph of the complete graph Kn is also known as the triangular graph, the Johnson graph J(n, 2), or the complement of the Kneser graph KGn,2. Triangular graphs are characterized by their spectra, except for n = 8.. See in particular Proposition 8, p. 262. They may also be characterized (again with the exception of K8) as the strongly regular graphs with parameters srg(n(n − 1)/2, 2(n − 2), n − 2, 4).
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The Apollonian networks do not form a family of graphs that is closed under the operation of taking graph minors, as removing edges but not vertices from an Apollonian network produces a graph that is not an Apollonian network. However, the planar partial 3-trees, subgraphs of Apollonian networks, are minor-closed. Therefore, according to the Robertson–Seymour theorem, they can be characterized by a finite number of forbidden minors. The minimal forbidden minors for the planar partial 3-trees are the four minimal graphs among the forbidden minors for the planar graphs and the partial 3-trees: the complete graph , the complete bipartite graph , the graph of the octahedron, and the graph of the pentagonal prism.
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For example, there are exactly 2 nodes in C that are mapped to the blue node in H. However, C is not a bipartite double cover of H or any other graph; it is not a bipartite graph. If we replace one triangle by a square in H the resulting graph has four distinct double covers. Two of them are bipartite but only one of them is the Kronecker cover. :Image:Covering-graph-4.svg As another example, the graph of the icosahedron is a double cover of the complete graph K6; to obtain a covering map from the icosahedron to K6, map each pair of opposite vertices of the icosahedron to a single vertex of K6.
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An arbitrary directed graph may also be transformed into a DAG, called its condensation, by contracting each of its strongly connected components into a single supervertex.. When the graph is already acyclic, its smallest feedback vertex sets and feedback arc sets are empty, and its condensation is the graph itself.
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In graph theory, the Kneser graph (alternatively ) is the graph whose vertices correspond to the -element subsets of a set of elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. Kneser graphs are named after Martin Kneser, who first investigated them in 1955.
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In mathematics, the bondage number of a nonempty graph is the cardinality of the smallest set E of edges such that the domination number of the graph with the edges E removed is strictly greater than the domination number of the original graph. The concept was introduced by Fink et. al.
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Thus, the game of GG played on the transformed graph will have the same outcome as on the original graph. This transformation takes time that is a constant multiple to the number of edge intersections in the original graph, thus it takes polynomial time. Thus planar GG is PSPACE-complete.
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Among the graphs that can be generated in this way, three of them are the Chang graphs. The Chang graphs are named after Chang Li-Chien, who proved that, with only these exceptions, every line graph of a complete graph is uniquely determined by its parameters as a strongly regular graph..
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The creation of the trajectory graph can be accomplished using k-nearest neighbors or minimum spanning tree algorithms. The topology of the trajectory refers to the structure of the graph and different algorithms are limited to creation of graph topologies of a particular type such as linear, branching, or cyclic.
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Perfectly orderable graphs are NP-complete to recognize.; . However, it is easy to test whether a particular ordering is a perfect ordering of a graph. Consequently, it is also NP-hard to find a perfect ordering of a graph, even if the graph is already known to be perfectly orderable.
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The Erdős–Stone theorem extends Turán's theorem by bounding the number of edges in a graph that does not have a fixed Turán graph as a subgraph. Via this theorem, similar bounds in extremal graph theory can be proven for any excluded subgraph, depending on the chromatic number of the subgraph.
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According to the Foster census, the Möbius–Kantor graph is the unique cubic symmetric graph with 16 vertices, and the smallest cubic symmetric graph which is not also distance-transitive.Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.
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Colin de Verdière is known for work in spectral theory, in particular on the semiclassical limit of quantum mechanics (including quantum chaos); in graph theory where he introduced a new graph invariant, the Colin de Verdière graph invariant; and on a variety of other subjects within Riemannian geometry and number theory.
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John Hopcroft and Robert Tarjan derived a means of testing the planarity of a graph in time linear to the number of edges. Their algorithm does this by constructing a graph embedding which they term a "palm tree". Efficient planarity testing is fundamental to graph drawing. Fan Chung et al.
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A connected component is a maximal connected subgraph of an undirected graph. Each vertex belongs to exactly one connected component, as does each edge. A graph is connected if and only if it has exactly one connected component. The strong components are the maximal strongly connected subgraphs of a directed graph.
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Usually the females of the family help the artisans to do this. Designing – The designs are made on the graph papers. One box on the graph denotes 6 yarns.
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It is NP-complete, given an -vertex cubic graph G and a parameter , to determine whether G can be obtained as a quotient of a planar graph with vertices..
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The Gray graph has chromatic number 2, chromatic index 3, radius 6 and diameter 6. It is also a 3-vertex-connected and 3-edge-connected non-planar graph.
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The Levi graphs of geometric configurations are biregular; a biregular graph is the Levi graph of an (abstract) configuration if and only if its girth is at least six..
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Some generalizations of the collaboration graph of mathematicians have also been considered. There is a hypergraph version,Frank Harary. Topics in Graph Theory. New York Academy of Sciences, 1979.
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