Mean first passage time is not symmetric, even for undirected graphs.
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Václav J. Havel is a Czech mathematician. He is known for characterizing the degree sequences of undirected graphs..
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Pancyclicity was first investigated in the context of tournaments by , , and . The concept of pancyclicity was named and extended to undirected graphs by .
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This is called the chromatic polynomial of our graph G (by analogy with the chromatic polynomial of undirected graphs) and can be denoted as \chi_G(k).
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Graphviz consists of a graph description language named the DOT languageThe DOT Language and a set of tools that can generate and/or process DOT files: ; dot : a command-line tool to produce layered drawings of directed graphs in a variety of output formats, such as (PostScript, PDF, SVG, annotated text and so on). ; neato : useful for undirected graphs. "spring model" layout, minimizes global energy. Useful for graphs up to about 1000 nodes ; fdp : useful for undirected graphs.
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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 Hanoi graph H^7_3 In graph theory and recreational mathematics, the Hanoi graphs are undirected graphs whose vertices represent the possible states of the Tower of Hanoi puzzle, and whose edges represent allowable moves between pairs of states.
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2-satisfiability has also been applied to problems of recognizing undirected graphs that can be partitioned into an independent set and a small number of complete bipartite subgraphs,. inferring business relationships among autonomous subsystems of the internet,. and reconstruction of evolutionary trees..
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Taking distances from or to all other nodes is irrelevant in undirected graphs, whereas it can produce totally different results in directed graphs (e.g. a website can have a high closeness centrality from outgoing link, but low closeness centrality from incoming links).
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The original proof was bijective and generalized the de Bruijn sequences. It is a variation on an earlier result by Smith and Tutte (1941). Counting the number of Eulerian circuits on undirected graphs is much more difficult. This problem is known to be #P-complete.
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The existence of a cycle in directed and undirected graphs can be determined by whether depth-first search (DFS) finds an edge that points to an ancestor of the current vertex (it contains a back edge).All the back edges which DFS skips over are part of cycles. In an undirected graph, the edge to the parent of a node should not be counted as a back edge, but finding any other already visited vertex will indicate a back edge. In the case of undirected graphs, only O(n) time is required to find a cycle in an n-vertex graph, since at most n − 1 edges can be tree edges.
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Other numbers defined in terms of edge deletion from undirected graphs include the edge connectivity, the minimum number of edges to delete in order to disconnect the graph, and matching preclusion, the minimum number of edges to delete in order to prevent the existence of a perfect matching.
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The price of stability was first studied by A. Schulzan and N. Moses and was so-called in the studies of E. Anshelevich. They showed that a pure strategy Nash equilibrium always exists and the price of stability of this game is at most the nth harmonic number in directed graphs. For undirected graphs Anshelevich and others presented a tight bound on the price of stability of 4/3 for a single source and two players case. Jian Li has proved that for undirected graphs with a distinguished destination to which all players must connect the price of stability of the Shapely network design game is O(\log n/\log\log n) where n is the number of players.
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This adjustment allows comparisons between nodes of graphs of different sizes. Taking distances from or to all other nodes is irrelevant in undirected graphs, whereas it can produce totally different results in directed graphs (e.g. a website can have a high closeness centrality from outgoing link, but low closeness centrality from incoming links).
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The windy postman problem is a variant of the route inspection problem in which the input is an undirected graph, but where each edge may have a different cost for traversing it in one direction than for traversing it in the other direction. In contrast to the solutions for directed and undirected graphs, it is NP-complete..
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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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A mixed graph is a graph in which some edges may be directed and some may be undirected. It is an ordered triple for a mixed simple graph and for a mixed multigraph with V, E (the undirected edges), A (the directed edges), ϕE and ϕA defined as above. Directed and undirected graphs are special cases.
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Paley graphs are dense undirected graphs, one for each prime p ≡ 1 (mod 4), that form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley digraphs are directed analogs of Paley graphs, one for each p ≡ 3 (mod 4), that yield antisymmetric conference matrices. The construction of these graphs uses quadratic residues.
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In particular, a complete graph with vertices, denoted , has no vertex cuts at all, but . A vertex cut for two vertices and is a set of vertices whose removal from the graph disconnects and . The local connectivity is the size of a smallest vertex cut separating and . Local connectivity is symmetric for undirected graphs; that is, .
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Suppose two directed or undirected graphs and with adjacency matrices and are given. and are isomorphic if and only if there exists a permutation matrix such that : P A_1 P^{-1} = A_2. In particular, and are similar and therefore have the same minimal polynomial, characteristic polynomial, eigenvalues, determinant and trace. These can therefore serve as isomorphism invariants of graphs.
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A signed digraph is a directed graph with signed arcs. Signed digraphs are far more complicated than signed graphs, because only the signs of directed cycles are significant. For instance, there are several definitions of balance, each of which is hard to characterize, in strong contrast with the situation for signed undirected graphs. Signed digraphs should not be confused with oriented signed graphs.
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There are variants of modular decomposition for undirected graphs and directed graphs. For each undirected graph, this decomposition is unique. This notion can be generalized to other structures (for example directed graphs) and is useful to design efficient algorithms for the recognition of some graph classes, for finding transitive orientations of comparability graphs, for optimization problems on graphs, and for graph drawing.
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To this end, GraphStream proposes several graph classes that allow to model directed and undirected graphs, 1-graphs or p-graphs (a.k.a. multigraphs, that are graphs that can have several edges between two nodes). GraphStream allows to store any kind of data attribute on the graph elements: numbers, strings, or any object. Moreover, in addition, GraphStream provides a way to handle the graph evolution in time.
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Johnson graphs are a special class of undirected graphs defined from systems of sets. The vertices of the Johnson graph J(n,k) are the k-element subsets of an n-element set; two vertices are adjacent when the intersection of the two vertices (subsets) contains (k-1)-elements.. Both Johnson graphs and the closely related Johnson scheme are named after Selmer M. Johnson.
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The Fibonacci cubes or Fibonacci networks are a family of undirected graphs with rich recursive properties derived from its origin in number theory. Mathematically they are similar to the hypercube graphs, but with a Fibonacci number of vertices. Fibonacci cubes were first explicitly defined in in the context of interconnection topologies for connecting parallel or distributed systems. They have also been applied in chemical graph theory.
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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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For simple undirected graphs, the first order theory of graphs includes the axioms :\forall u\bigl(\lnot(u\sim u)\bigr) (the graph cannot contain any loops), and :\forall u\forall v(u\sim v\Rightarrow v\sim u) (edges are undirected). Other types of graphs, such as directed graphs, may involve different axioms, and logical formulations of multigraph properties require having separate variables for vertices and edges.
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In model theory, a graph is just a structure. But in that case, there is no limitation on the number of edges: it can be any cardinal number, see continuous graph. In computational biology, power graph analysis introduces power graphs as an alternative representation of undirected graphs. In geographic information systems, geometric networks are closely modeled after graphs, and borrow many concepts from graph theory to perform spatial analysis on road networks or utility grids.
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PPAD is contained in (but not known to be equal to) PPA (the corresponding class of parity arguments for undirected graphs) which is contained in TFNP. PPAD is also contained in (but not known to be equal to) PPP, another subclass of TFNP. It contains CLS. PPAD is a class of problems that are believed to be hard, but obtaining PPAD-completeness is a weaker evidence of intractability than that of obtaining NP-completeness.
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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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Formally, let be any graph, and let be any subset of vertices of . Then the induced subgraph is the graph whose vertex set is and whose edge set consists of all of the edges in that have both endpoints in .. The same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph may also be called the subgraph induced in by , or (if context makes the choice of unambiguous) the induced subgraph of .
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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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One of his best-known results is a linear-time algorithm for the single-source shortest paths problem in undirected graphs (Thorup, 1999).Robbins Prize Citation With Mihai Pătraşcu he has shown that simple tabulation hashing schemes achieve the same or similar performance criteria as hash families that have higher independence in worst case, while permitting speedier implementations.Regan, Tabulation hashing and independence, Gödel’s Lost Letter, April 14, 2012, Fortnow, Complexity year in review, December 29, 2011.
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In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally. Paley graphs are named after Raymond Paley.
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Savitch's theorem guarantees that the algorithm can be simulated in O(log2 n) deterministic space. The same problem for undirected graphs is called undirected s-t connectivity and was shown to be L-complete by Omer Reingold. This research won him the 2005 Grace Murray Hopper Award. Undirected st- connectivity was previously known to be complete for the class SL, so Reingold's work showed that SL is the same class as L. On alternating graphs, the problem is P-complete .
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"spring model" which minimizes forces instead of energy ; sfdp : multiscale version of fdp for the layout of large undirected graphs ; twopi : for radial graph layouts. Nodes are placed on concentric circles depending their distance from a given root node ; circo : circular layout. Suitable for certain diagrams of multiple cyclic structures, such as certain telecommunications networks ; dotty : a graphical user interface to visualize and edit graphs. ; lefty : a programmable (in a language inspired by EZThe Lefty guide (“Editing Pictures with lefty”), section 3.1, p.
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Another interesting connection concerns orientations of graphs. An orientation of an undirected graph G is any directed graph obtained by choosing one of the two possible orientations for each edge. An example of an orientation of the complete graph Kk is the transitive tournament k with vertices 1,2,…,k and arcs from i to j whenever i < j. A homomorphism between orientations of graphs G and H yields a homomorphism between the undirected graphs G and H, simply by disregarding the orientations.
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It is distinct from the order → on equivalence classes of undirected graphs, but contains it as a suborder. This is because every undirected graph can be thought of as a directed graph where every arc (u,v) appears together with its inverse arc (v,u), and this does not change the definition of homomorphism. The order → for directed graphs is again a distributive lattice and a Heyting algebra, with join and meet operations defined as before. However, it is not dense.
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The undirected problem is APX-complete, which directly follows from the APX-completeness of the vertex cover problem,, the existence of an approximation preserving L-reduction from the vertex cover problem to it and existing approximation algorithms. The best known approximation algorithm on undirected graphs is by a factor of two.. See also for an alternative approximation algorithm with the same approximation ratio. Whether the directed version is polynomial time approximable within constant ratio and thereby APX-complete is an open question.
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A graphoid is termed DAG-induced if there exists a directed acyclic graph D such that I(X,Z,Y) \Leftrightarrow \langle X,Z,Y\rangle_D where \langle X,Z,Y\rangle_D stands for d-separation in D. d-separation (d-connotes "directional") extends the notion of vertex separation from undirected graphs to directed acyclic graphs. It permits the reading of conditional independencies from the structure of Bayesian networks. However, conditional independencies in a DAG cannot be completely characterized by a finite set of axioms.
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The cycle rank is an invariant of directed graphs that measures the level of nesting of cycles in the graph. It has a more complicated definition than circuit rank (closely related to the definition of tree-depth for undirected graphs) and is more difficult to compute. Another problem for directed graphs related to the circuit rank is the minimum feedback arc set, the smallest set of edges whose removal breaks all directed cycles. Both cycle rank and the minimum feedback arc set are NP-hard to compute.
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Ram Prakash Gupta was a professor of graph theory at Waterloo, Canada and at Ohio State University. He received his Ph.D. in graph theory from the Indian Statistical Institute, Calcutta, India in 1968; his official advisor was C. R. Rao, but he did much of his doctoral work under the mentorship of S. S. Shrikhande. Gupta is known for his independent discovery of Vizing's theorem on edge coloring of undirected graphs, which he announced two years after Vizing's Russian-language publication of the theorem.
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The tournament has one vertex for each color in the coloring. For each pair of colors, there is an edge in the colored graph with those two colors at its endpoints, which lends its orientation to the edge in the tournament between the vertices corresponding to the two colors. Incomplete colorings may also be represented by homomorphisms into tournaments but in this case the correspondence between colorings and homomorphisms is not one-to-one. Undirected graphs of bounded genus, bounded degree, or bounded acyclic chromatic number also have bounded oriented chromatic number.
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Modularity is the fraction of the edges that fall within the given groups minus the expected fraction if edges were distributed at random. The value of the modularity for unweighted and undirected graphs lies in the range [-1/2,1]. It is positive if the number of edges within groups exceeds the number expected on the basis of chance. For a given division of the network's vertices into some modules, modularity reflects the concentration of edges within modules compared with random distribution of links between all nodes regardless of modules.
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Several algorithms exist to find shortest and longest paths in graphs, with the important distinction that the former problem is computationally much easier than the latter. Dijkstra's algorithm produces a list of shortest paths from a source vertex to every other vertex in directed and undirected graphs with non-negative edge weights (or no edge weights), whilst the Bellman–Ford algorithm can be applied to directed graphs with negative edge weights. The Floyd–Warshall algorithm can be used to find the shortest paths between all pairs of vertices in weighted directed graphs.
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The algorithm suggested by Gallager, Humblet, and Spira for general undirected graphs has had a strong impact on the design of distributed algorithms in general, and won the Dijkstra Prize for an influential paper in distributed computing. Many other algorithms have been suggested for different kinds of network graphs, such as undirected rings, unidirectional rings, complete graphs, grids, directed Euler graphs, and others. A general method that decouples the issue of the graph family from the design of the leader election algorithm was suggested by Korach, Kutten, and Moran.
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In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity. Formally, an automorphism of a graph G = (V,E) is a permutation σ of the vertex set V, such that the pair of vertices (u,v) form an edge if and only if the pair (σ(u),σ(v)) also form an edge. That is, it is a graph isomorphism from G to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs.
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Judea Pearl and Azaria Paz coined the term "graphoids" after discovering that a set of axioms that govern conditional independence in probability theory is shared by undirected graphs. Variables are represented as nodes in a graph in such a way that variable sets X and Y are independent conditioned on Z in the distribution whenever node set Z separates X from Y in the graph. Axioms for conditional independence in probability were derived earlier by A. Philip Dawid and Wolfgang Spohn. The correspondence between dependence and graphs was later extended to directed acyclic graphs (DAGs) and to other models of dependency.
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A graph may be fully specified by its adjacency matrix A, which is an nxn square matrix, with Aij specifying the nature of the connection between vertex i and vertex j. For a simple graph, Aij= 0 or 1, indicating disconnection or connection respectively, with Aii=0. Graphs with self-loops will be characterized by some or all Aii being equal to a positive integer, and multigraphs (with multiple edges between vertices) will be characterized by some or all Aij being equal to a positive integer. Undirected graphs will have a symmetric adjacency matrix (Aij=Aji).
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The algorithm suggested by Gallager, Humblet, and Spira for general undirected graphs has had a strong impact on the design of distributed algorithms in general, and won the Dijkstra Prize for an influential paper in distributed computing. Many other algorithms were suggested for different kind of network graphs, such as undirected rings, unidirectional rings, complete graphs, grids, directed Euler graphs, and others. A general method that decouples the issue of the graph family from the design of the coordinator election algorithm was suggested by Korach, Kutten, and Moran. In order to perform coordination, distributed systems employ the concept of coordinators.
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Seifollah Louis Hakimi (1932 - June 23, 2005) was an Iranian-American mathematician born in Iran, a professor emeritus at Northwestern University, where he chaired the department of electrical engineering from 1973 to 1978.. He was Chair of the Department of Electrical Engineering at University of California, Davis, from 1986 to 1996. Hakimi received his Ph.D. from the University of Illinois at Urbana-Champaign in 1959, under the supervision of Mac Van Valkenburg. He has over 100 academic descendants, most of them via his student Narsingh Deo. He is known for characterizing the degree sequences of undirected graphs,.
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JUNG's architecture is designed to support a variety of representations of entities and their relations, such as directed and undirected graphs, , graphs with parallel edges, and hypergraphs. It provides a mechanism for annotating graphs, entities, and relations with metadata. JUNG also facilitates the creation of analytic tools for complex data sets that can examine the relations between entities as well as the metadata attached to each entity and relation. JUNG includes implementations of a number of algorithms from graph theory, data mining, and social network analysis, such as routines for clustering, , , random graph generation, statistical analysis, and calculation of network distances, flows, and importance measures.
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On the other hand, given a homomorphism G → H between undirected graphs, any orientation of H can be pulled back to an orientation of G so that has a homomorphism to . Therefore, a graph G is k-colorable (has a homomorphism to Kk) if and only if some orientation of G has a homomorphism to k. A folklore theorem states that for all k, a directed graph G has a homomorphism to k if and only if it admits no homomorphism from the directed path k+1. Here n is the directed graph with vertices 1, 2, …, n and edges from i to i + 1, for i = 1, 2, …, n − 1.
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More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices. Homomorphisms generalize various notions of graph colorings and allow the expression of an important class of constraint satisfaction problems, such as certain scheduling or frequency assignment problems. The fact that homomorphisms can be composed leads to rich algebraic structures: a preorder on graphs, a distributive lattice, and a category (one for undirected graphs and one for directed graphs). The computational complexity of finding a homomorphism between given graphs is prohibitive in general, but a lot is known about special cases that are solvable in polynomial time.
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In graph theory, Vizing's theorem states that every simple undirected graph may be edge colored using a number of colors that is at most one larger than the maximum degree of the graph. At least colors are always necessary, so the undirected graphs may be partitioned into two classes: "class one" graphs for which colors suffice, and "class two" graphs for which colors are necessary. A more generalized version of Vizing's theorem states that every undirected multigraph without loops can be colored with at most colors, where is the multiplicity of the multigraph. The theorem is named for Vadim G. Vizing who published it in 1964.
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In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the (simple) 3-vertex- connected planar graphs (with at least four vertices). That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. Branko Grünbaum has called this theorem “the most important and deepest known result on 3-polytopes.” The theorem appears in a 1922 paper of Ernst Steinitz, after whom it is named.
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Hamiltonicity can be expressed in MSO2 by the existence of a set of edges that forms a connected 2-regular graph on all the vertices, with connectivity expressed as above and 2-regularity expressed as the incidence of two but not three distinct edges at each vertex. However, Hamiltonicity is not expressible in MSO1, because MSO1 is not capable of distinguishing complete bipartite graphs with equal numbers of vertices on each side of the bipartition (which are Hamiltonian) from unbalanced complete bipartite graphs (which are not).; , Corollary 7.24, pp. 126–127. Although not part of the definition of MSO2, orientations of undirected graphs can be represented by a technique involving Trémaux trees.
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Pavol Hell and Jaroslav Nešetřil proved that, for undirected graphs, no other case is tractable: : Hell–Nešetřil theorem (1990): The H-coloring problem is in P when H is bipartite and NP-complete otherwise. This is also known as the dichotomy theorem for (undirected) graph homomorphisms, since it divides H-coloring problems into NP-complete or P problems, with no intermediate cases. For directed graphs, the situation is more complicated and in fact equivalent to the much more general question of characterizing the complexity of constraint satisfaction problems. It turns out that H-coloring problems for directed graphs are just as general and as diverse as CSPs with any other kinds of constraints.
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In this article, unless stated otherwise, graphs are finite, undirected graphs with loops allowed, but multiple edges (parallel edges) disallowed. A graph homomorphismFor introductions, see (in order of increasing length): ; ; . f from a graph G = (V(G), E(G)) to a graph H = (V(H), E(H)), written : is a function from V(G) to V(H) that maps endpoints of each edge in G to endpoints of an edge in H. Formally, {u,v} ∈ E(G) implies {f(u),f(v)} ∈ E(H), for all pairs of vertices u, v in V(G). If there exists any homomorphism from G to H, then G is said to be homomorphic to H or H-colorable.
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In constructing matchings in undirected graphs, it is important to find alternating paths, paths of vertices that start and end at unmatched vertices, in which the edges at odd positions in the path are not part of a given partial matching and in which the edges at even positions in the path are part of the matching. By removing the matched edges of such a path from a matching, and adding the unmatched edges, one can increase the size of the matching. Similarly, cycles that alternate between matched and unmatched edges are of importance in weighted matching problems. As showed, an alternating path or cycle in an undirected graph may be modeled as a regular path or cycle in a skew-symmetric directed graph.
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If a graph has a Hamiltonian path, then that path (rooted at one of its endpoints) is also a Trémaux tree. The undirected graphs for which every Trémaux tree has this form are the cycle graphs, complete graphs, and balanced complete bipartite graphs.. Trémaux trees are closely related to the concept of tree-depth. The tree-depth of a graph G can be defined as the smallest number d such that G can be embedded as a subgraph of a graph H that has a Trémaux tree T of depth d. Bounded tree- depth, in a family of graphs, is equivalent to the existence of a path that cannot occur as a graph minor of the graphs in the family.
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In the mathematical field of graph theory, the Chang graphs are a set of three 12-regular undirected graphs, each with 28 vertices and 168 edges. They are strongly regular, with the same parameters and spectra as the line graph L(K8) of the complete graph K8. Each of these three graphs may be obtained by graph switching from L(K8). That is, a subset S of the vertices of L(K8) is chosen, each edge that connects a vertex in S with a vertex not in S is deleted from L(K8), and an edge is added for each pair of vertices (with again one in S and one not in S) that were not already connected by an edge.
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In the mathematical theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset S of elements of the matroid is, similarly, the maximum size of an independent subset of S, and the rank function of the matroid maps sets of elements to their ranks. The rank function is one of the fundamental concepts of matroid theory via which matroids may be axiomatized. The rank functions of matroids form an important subclass of the submodular set functions, and the rank functions of the matroids defined from certain other types of mathematical object such as undirected graphs, matrices, and field extensions are important within the study of those objects.
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Transitive tournaments play a role in Ramsey theory analogous to that of cliques in undirected graphs. In particular, every tournament on n vertices contains a transitive subtournament on 1+\lfloor\log_2 n\rfloor vertices.. The proof is simple: choose any one vertex v to be part of this subtournament, and form the rest of the subtournament recursively on either the set of incoming neighbors of v or the set of outgoing neighbors of v, whichever is larger. For instance, every tournament on seven vertices contains a three-vertex transitive subtournament; the Paley tournament on seven vertices shows that this is the most that can be guaranteed . However, showed that this bound is not tight for some larger values of n.
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The pathwidth of an arbitrary undirected graph G may be defined as the smallest number w such that there exists an interval graph H containing G as a subgraph, with the largest clique in H having w + 1 vertices. For trees (viewed as undirected graphs by forgetting their orientation and root) the pathwidth differs from the Strahler number, but is closely related to it: in a tree with pathwidth w and Strahler number s, these two numbers are related by the inequalities, using a definition of the "dimension" of a tree that is one less than the Strahler number. :w ≤ s ≤ 2w + 2. The ability to handle graphs with cycles and not just trees gives pathwidth extra versatility compared to the Strahler number.
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Applying the Hajós construction to two copies of by identifying a vertex from each copy into a single vertex (shown with both colors), deleting an edge incident to the combined vertex within each subgraph (dashed) and adding a new edge connecting the endpoints of the deleted edges (thick green), produces the Moser spindle. Let and be two undirected graphs, be an edge of , and be an edge of . Then the Hajós construction forms a new graph that combines the two graphs by identifying vertices and into a single vertex, removing the two edges and , and adding a new edge . For example, let and each be a complete graph on four vertices; because of the symmetry of these graphs, the choice of which edge to select from each of them is unimportant.
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In contrast with ordinary undirected graphs for which there is a single natural notion of cycles and acyclic graphs, there are multiple natural non-equivalent definitions of acyclicity for hypergraphs which collapse to ordinary graph acyclicity for the special case of ordinary graphs. A first definition of acyclicity for hypergraphs was given by Claude Berge: a hypergraph is Berge- acyclic if its incidence graph (the bipartite graph defined above) is acyclic. This definition is very restrictive: for instance, if a hypergraph has some pair v eq v' of vertices and some pair f eq f' of hyperedges such that v, v' \in f and v, v' \in f', then it is Berge-cyclic. Berge-cyclicity can obviously be tested in linear time by an exploration of the incidence graph.
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The betweenness may be normalised by dividing through the number of pairs of vertices not including v, which for directed graphs is (n-1)(n-2) and for undirected graphs is (n-1)(n-2)/2. For example, in an undirected star graph, the center vertex (which is contained in every possible shortest path) would have a betweenness of (n-1)(n-2)/2 (1, if normalised) while the leaves (which are contained in no shortest paths) would have a betweenness of 0. From a calculation aspect, both betweenness and closeness centralities of all vertices in a graph involve calculating the shortest paths between all pairs of vertices on a graph, which requires O(V^3) time with the Floyd–Warshall algorithm. However, on sparse graphs, Johnson's algorithm may be more efficient, taking O(V^2 \log V + V E) time.
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Generalized to directed graphs, the conjecture has simple counterexamples, as observed by . Here, the chromatic number of a directed graph is just the chromatic number of the underlying graph, but the tensor product has exactly half the number of edges (for directed edges g→g' in G and h→h' in H, the tensor product G × H has only one edge, from (g,h) to (g',h'), while the product of the underlying undirected graphs would have an edge between (g,h') and (g',h) as well). However, the Weak Hedetniemi Conjecture turns out to be equivalent in the directed and undirected settings . The problem cannot be generalized to infinite graphs: gave an example of two infinite graphs, each requiring an uncountable number of colors, such that their product can be colored with only countably many colors.
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One formulation of the conjecture involves embeddings of the shortest path distances of weighted undirected graphs into \ell_1 spaces, real vector spaces in which the distance between two vectors is the sum of their coordinate differences. If an embedding maps all pairs of vertices with distance d to pairs of vectors with distance in the range [cd,Cd] then its stretch factor or distortion is the ratio C/c; an isometry has stretch factor one, and all other embeddings have greater stretch factor. The graphs that have an embedding with at most a given distortion are closed under graph minor operations, operations that delete vertices or edges from a graph or contract some of its edges. The GNRS conjecture states that, conversely, every nontrivial minor-closed family of graphs can be embedded into an \ell_1 space with bounded distortion.
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NP-completeness reduction from 3-satisfiability to graph 3-coloring. The gadgets for variables and clauses are shown on the upper and lower left, respectively; on the right is an example of the entire reduction for the 3-CNF formula with three variables and two clauses. Many NP- completeness proofs are based on many-one reductions from 3-satisfiability, the problem of finding a satisfying assignment to a Boolean formula that is a conjunction (Boolean and) of clauses, each clause being the disjunction (Boolean or) of three terms, and each term being a Boolean variable or its negation. A reduction from this problem to a hard problem on undirected graphs, such as the Hamiltonian cycle problem or graph coloring, would typically be based on gadgets in the form of subgraphs that simulate the behavior of the variables and clauses of a given 3-satisfiability instance.
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A modular graph derived from a modular lattice In graph theory, a branch of mathematics, the modular graphs are undirected graphs in which every three vertices , , and have at least one median vertex that belongs to shortest paths between each pair of , , and .Modular graphs, Information System on Graph Classes and their Inclusions, retrieved 2016-09-30. Their name comes from the fact that a finite lattice is a modular lattice if and only if its Hasse diagram is a modular graph.. It is not possible for a modular graph to contain a cycle of odd length. For, if is a shortest odd cycle in a graph, is a vertex of , and is the edge of the cycle farthest from , there could be no median , for the only vertices on the shortest path are and themselves, but neither can belong to a shortest path from to the other without shortcutting and creating a shorter odd cycle.
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For any partition of the vertices of the Rado graph into two sets A and B, or more generally for any partition into finitely many subsets, at least one of the subgraphs induced by one of the partition sets is isomorphic to the whole Rado graph. gives the following short proof: if none of the parts induces a subgraph isomorphic to the Rado graph, they all fail to have the extension property, and one can find pairs of sets Ui and Vi that cannot be extended within each subgraph. But then, the union of the sets Ui and the union of the sets Vi would form a set that could not be extended in the whole graph, contradicting the Rado graph's extension property. This property of being isomorphic to one of the induced subgraphs of any partition is held by only three countably infinite undirected graphs: the Rado graph, the complete graph, and the empty graph.
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The relation → is a partial order on those equivalence classes; it defines a poset. Let G < H denote that there is a homomorphism from G to H, but no homomorphism from H to G. The relation → is a dense order, meaning that for all (undirected) graphs G, H such that G < H, there is a graph K such that G < K < H (this holds except for the trivial cases G = K0 or K1). For example, between any two complete graphs (except K0, K1) there are infinitely many circular complete graphs, corresponding to rational numbers between natural numbers. The poset of equivalence classes of graphs under homomorphisms is a distributive lattice, with the join of [G] and [H] defined as (the equivalence class of) the disjoint union [G ∪ H], and the meet of [G] and [H] defined as the tensor product [G × H] (the choice of graphs G and H representing the equivalence classes [G] and [H] does not matter).
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It is straightforward to verify that the graph minor relation forms a partial order on the isomorphism classes of undirected graphs: it is transitive (a minor of a minor of G is a minor of G itself), and G and H can only be minors of each other if they are isomorphic because any nontrivial minor operation removes edges or vertices. A deep result by Neil Robertson and Paul Seymour states that this partial order is actually a well-quasi-ordering: if an infinite list G1, G2,... of finite graphs is given, then there always exist two indices i < j such that Gi is a minor of Gj. Another equivalent way of stating this is that any set of graphs can have only a finite number of minimal elements under the minor ordering., Chapter 12: Minors, Trees, and WQO; . This result proved a conjecture formerly known as Wagner's conjecture, after Klaus Wagner; Wagner had conjectured it long earlier, but only published it in 1970.
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For a simple graph with vertex set , the adjacency matrix is a square matrix such that its element is one when there is an edge from vertex to vertex , and zero when there is no edge.. The diagonal elements of the matrix are all zero, since edges from a vertex to itself (loops) are not allowed in simple graphs. It is also sometimes useful in algebraic graph theory to replace the nonzero elements with algebraic variables.. The same concept can be extended to multigraphs and graphs with loops by storing the number of edges between each two vertices in the corresponding matrix element, and by allowing nonzero diagonal elements. Loops may be counted either once (as a single edge) or twice (as two vertex-edge incidences), as long as a consistent convention is followed. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention.
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The following pseudocode shows IDDFS implemented in terms of a recursive depth-limited DFS (called DLS) for directed graphs. This implementation of IDDFS does not account for already-visited nodes and therefore does not work for undirected graphs. function IDDFS(root) is for depth from 0 to ∞ do found, remaining ← DLS(root, depth) if found ≠ null then return found else if not remaining then return null function DLS(node, depth) is if depth = 0 then if node is a goal then return (node, true) else return (null, true) (Not found, but may have children) else if depth > 0 then any_remaining ← false foreach child of node do found, remaining ← DLS(child, depth−1) if found ≠ null then return (found, true) if remaining then any_remaining ← true (At least one node found at depth, let IDDFS deepen) return (null, any_remaining) If the goal node is found, then DLS unwinds the recursion returning with no further iterations. Otherwise, if at least one node exists at that level of depth, the remaining flag will let IDDFS continue.
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In graph theory, the Robertson–Seymour theorem (also called the graph minor theorem.) states that the undirected graphs, partially ordered by the graph minor relationship, form a well-quasi-ordering.. Equivalently, every family of graphs that is closed under minors can be defined by a finite set of forbidden minors, in the same way that Wagner's theorem characterizes the planar graphs as being the graphs that do not have the complete graph K5 or the complete bipartite graph K3,3 as minors. The Robertson–Seymour theorem is named after mathematicians Neil Robertson and Paul D. Seymour, who proved it in a series of twenty papers spanning over 500 pages from 1983 to 2004.; . Before its proof, the statement of the theorem was known as Wagner's conjecture after the German mathematician Klaus Wagner, although Wagner said he never conjectured it.. A weaker result for trees is implied by Kruskal's tree theorem, which was conjectured in 1937 by Andrew Vázsonyi and proved in 1960 independently by Joseph Kruskal and S. Tarkowski.
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A 2-simplex, for instance, can be thought of as a two-dimensional "triangular" shape bounded by a list of three vertices A, B, C and three arrows B -> C, A -> C and A -> B. In general, an n-simplex is an object made up from a list of n + 1 vertices (which are 0-simplices) and n + 1 faces (which are (n − 1)-simplices). The vertices of the i-th face are the vertices of the n-simplex minus the i-th vertex. The vertices of a simplex need not be distinct and a simplex is not determined by its vertices and faces: two different simplices may share the same list of faces (and therefore the same list of vertices), just like two different arrows in a multigraph may connect the same two vertices. Simplicial sets should not be confused with abstract simplicial complexes, which generalize simple undirected graphs rather than directed multigraphs. Formally, a simplicial set X is a collection of sets Xn, n = 0, 1, 2, ..., together with certain maps between these sets: the face maps dn,i : Xn -> Xn−1 (n = 1, 2, 3, ... and 0 ≤ i ≤ n) and degeneracy maps sn,i : Xn->Xn+1 (n = 0, 1, 2, ... and 0 ≤ i ≤ n).
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