599 Sentences With "undirected" | Random Sentence Generator

Do you think there is a connection between daily undirected play and creativity?

You can get up, it's not a Broadway show, everything is very undirected.

That said, it's a mess of a book, fuzzy, disorganized, and maddeningly undirected.

A standout work in this exhibition is the stop-motion film, Delights of an Undirected Mind (2016).

The rewards are far and few in between, making it almost impossible for undirected exploration schemes to succeed.

And this I would fight for: the freedom of the mind to take any direction it wishes, undirected.

If the links between nodes are bidirectional, we say that the graph is undirected; otherwise, it's a directed graph.

What I wasn't expecting was for Homecoming to by catalyzed by a dangerous and undirected working-class rage and fear.

A movement is arising, undirected and driven largely by students, to scrub campuses clean of words, ideas, and subjects that might cause discomfort or give offense.

And yet the rage that one felt was an abstract, undirected emotion which could be switched from one object to another like the flame of a blowlamp.

We conclude that CTVT has really ceased to evolve in any particular 'direction,' and so its evolution is largely undirected or, if you like, 'random' (this is known as 'genetic drift').

Delights of an Undirected Mind offers a counterpoint to Monsters & Myths with oftentimes playful, whimsical but no less visceral imagery and experiences that push the visitors out of their comfort zones.

But Cheng presents abstractions of a world that is in fact not very different from our own — a fantastical, well-orchestrated drama of our own flailing, undirected, cannibalistic bodies and existential dilemmas.

On display at the Baltimore Museum of Art (BMA) are two concurrent exhibitions, Monsters & Myths: Surrealism and War in the 1930s and 1940s and Nathalie Djurberg and Hans Berg/Delights of an Undirected Mind.

Under research, principles included the need to create "beneficial intelligence" as opposed to "undirected intelligence" (more on this in just a bit), and an admonition for AI developers to maintain a healthy dialogue with policy-makers.

The singers seemed genuinely undirected, lost in front of the set's imposing mass; the production team was scrambling to make the effects work, but it appeared that relatively little attention had been given to the acting.

The Artists Pick Artists series was designed to take readers — along with me and Hyperallergic — on an undirected journey through the art world by artists, on their own terms, collectively and individually, without an end in sight.

It's funny to try to avoid feeling self-consciousness because that's a goal, to be conscious of yourself, but the other kind of self-consciousness, if it's undirected, it can be really debilitating and make you socially withdrawn.

" He writes that the latter half of the 19th century was "an incredible era of violence, greed, audacity, sentimentality, undirected exuberance, and an almost reverential attitude toward the ideal of personal freedom for those who already had it.

Eventually that milk will be spread evenly throughout your coffee, and these cells are doing the same sort of thing - they're moving in an undirected manner and eventually, slowly, they manage to fully colonize the skin of this animal.

So Ms. Dathan, who was elected to the Connecticut legislature in November, agreed to support a state bill that would require schools to provide at least 50 minutes of daily undirected play for students enrolled in preschool through fifth grade.

Blocks of scarlet behind the head and branches articulate a geometric space that seems at first absent from the visual maelstrom of Oehlen's earlier pieces, but in "Frau im Baum II" and much of the work on view, what may at first look like random, undirected energy and forms unfolds into a compelling interplay of chaos and control.

Monsters & Myths: Surrealism and War in the 1930s and 1940s, curated by Oliver Shell, BMA Associate Curator of European Art, and Oliver Tostmann, Susan Morse Hilles Curator of European Art at the Wadsworth Atheneum Museum of Art, and Nathalie Djurberg and Hans Berg/Delights of an Undirected Mind, curated by Laura Albans, Assistant Curator of European Painting and Sculpture, both continue at the Baltimore Museum of Art through May 26.

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.

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.

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.

One may also consider playing either Geography game on an undirected graph (that is, the edges can be traversed in both directions). Fraenkel, Scheinerman, and Ullman show that undirected vertex geography can be solved in polynomial time, whereas undirected edge geography is PSPACE- complete, even for planar graphs with maximum degree 3. If the graph is bipartite, then Undirected Edge Geography is solvable in polynomial time.

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.

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.

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.

Mean first passage time is not symmetric, even for undirected graphs.

In the undirected edge- disjoint paths problem, we are given an undirected graph and two vertices and , and we have to find the maximum number of edge-disjoint s-t paths in . The Menger's theorem states that the maximum number of edge-disjoint s-t paths in an undirected graph is equal to the minimum number of edges in an s-t cut-set.

Similar ideas may be applied to undirected, and possibly cyclic, graphs such as Markov networks.

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. Some authors restrict the phrase "directed forest" to the case where the edges of each connected component are all directed towards a particular vertex, or all directed away from a particular vertex (see branching).

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.

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.

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. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic. Some authors restrict the phrase "directed tree" to the case where the edges are all directed towards a particular vertex, or all directed away from a particular vertex (see arborescence).

Mega-merger is a distributed algorithm aimed at solving the election problem in generic connected undirected graph.

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 .

Václav J. Havel is a Czech mathematician. He is known for characterizing the degree sequences of undirected graphs..

If is the adjacency matrix of the directed or undirected graph , then the matrix (i.e., the matrix product of copies of ) has an interesting interpretation: the element gives the number of (directed or undirected) walks of length from vertex to vertex . If is the smallest nonnegative integer, such that for some , , the element of is positive, then is the distance between vertex and vertex . This implies, for example, that the number of triangles in an undirected graph is exactly the trace of divided by 6.

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.

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.

André Joyal used this fact to provide a bijective proof of Cayley's formula, that the number of undirected trees on n nodes is nn − 2, by finding a bijection between maximal directed pseudoforests and undirected trees with two distinguished nodes.. If self-loops are not allowed, the number of maximal directed pseudoforests is instead (n − 1)n.

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..

A symmetric sparse matrix arises as the adjacency matrix of an undirected graph; it can be stored efficiently as an adjacency list.

Pancyclicity was first investigated in the context of tournaments by , , and . The concept of pancyclicity was named and extended to undirected graphs by .

In graph theory, a connected dominating set and a maximum leaf spanning tree are two closely related structures defined on an undirected graph.

An incomparability graph is an undirected graph that connects pairs of elements that are not comparable to each other in a partial order.

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.

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.

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.

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.

A mathematical problem is wild if it contains the problem of classifying pairs of square matrices up to simultaneous similarity. Examples of wild problems are classifying indecomposable representations of any quiver that is neither a Dynkin quiver (i.e. the underlying undirected graph of the quiver is a (finite) Dynkin diagram) nor a Euclidean quiver (i.e., the underlying undirected graph of the quiver is an affine Dynkin diagram).

In computational complexity theory, SL (Symmetric Logspace or Sym-L) is the complexity class of problems log-space reducible to USTCON (undirected s-t connectivity), which is the problem of determining whether there exists a path between two vertices in an undirected graph, otherwise described as the problem of determining whether two vertices are in the same connected component. This problem is also called the undirected reachability problem. It does not matter whether many-one reducibility or Turing reducibility is used. Although originally described in terms of symmetric Turing machines, that equivalent formulation is very complex, and the reducibility definition is what is used in practice.

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.

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.

To emphasize that DAGs are not the same thing as directed versions of undirected acyclic graphs, some authors call them acyclic directed graphs or acyclic digraphs.

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).

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.

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.

This is the definition used, e.g., by . Equivalently, it is an undirected graph in which each connected component has no more edges than vertices.This is the definition in .

In 2005 Omer Reingold introduced an algorithm that solves the undirected st-connectivity problem, the problem of testing whether there is a path between two given vertices in an undirected graph, using only logarithmic space. The algorithm relies heavily on the zigzag product. Roughly speaking, in order to solve the undirected s-t connectivity problem in logarithmic space, the input graph is transformed, using a combination of powering and the zigzag product, into a constant-degree regular graph with a logarithmic diameter. The power product increases the expansion (hence reduces the diameter) at the price of increasing the degree, and the zigzag product is used to reduce the degree while preserving the expansion.

This proof is inspired by . Let be a simple undirected graph. We proceed by induction on , the number of edges. If the graph is empty, the theorem trivially holds.

Yo-Yo is a distributed algorithm aimed at minimum finding and leader election in generic connected undirected graph. Unlike Mega-Merger it has a trivial termination and cost analysis.

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 }.

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.

Sexual harassment in the military includes a broad spectrum of behaviour. Undirected behaviours that affect the working environment, such as sexist jokes and the prominent display of pornographic material, may constitute sexual harassment, as do directed behaviours targeted at one or more individuals, such as unwanted sexual advances and sexual assault. Research in Canada has found that a culture of undirected sexual harassment increases the risk of directed sexual harassment and assault.

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.

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.

A symmetric Turing machine is a Turing machine which has a configuration graph that is undirected (that is, configuration i yields configuration j if and only if j yields i).

Replicated softmax: an undirected topic model. Neural Information Processing Systems 23. and even many body quantum mechanics. They can be trained in either supervised or unsupervised ways, depending on the task.

A graph is connected if every vertex or edge is reachable from every other vertex or edge. A cycle in an undirected graph is a connected subgraph in which each vertex is incident to exactly two edges, or is a loop.See the linked articles and the references therein for these definitions. The 21 unicyclic graphs with at most six vertices A pseudoforest is an undirected graph in which each connected component contains at most one cycle.

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.

In a directed graph, an ordered pair of vertices is called strongly connected if a directed path leads from x to y. Otherwise, the ordered pair is called weakly connected if an undirected path leads from x to y after replacing all of its directed edges with undirected edges. Otherwise, the ordered pair is called disconnected. A strongly connected graph is a directed graph in which every ordered pair of vertices in the graph is strongly connected.

Typically, a family of universality classes will have a lower and upper critical dimension: below the lower critical dimension, the universality class becomes degenerate (this dimension is 2d for the Ising model, or for directed percolation, but 1d for undirected percolation), and above the upper critical dimension the critical exponents stabilize and can be calculated by an analog of mean field theory (this dimension is 4d for Ising or for directed percolation, and 6d for undirected percolation).

Under EAC there is an assumption of a positive selection pressure driving evolution after gene duplication, whereas the DDC model only requires neutral ("undirected") evolution to take place, i.e. degeneration and complementation.

In the mathematical subfield of graph theory a level structure of an undirected graph is a partition of the vertices into subsets that have the same distance from a given root vertex..

Emotions elicited by listening to music are another potential example of undirected, nonintentional emotions. Emotions aroused in this way do not seem to necessarily be about anything, including the music that arouses them.

"Undirected calls" are used for stations in transit, and Payton uses this to communicate with Grayman without giving away his position. While he uses a directed signal, he requires Grayman to send undirected. This is crucial to the plot, for Allen uses a radio direction finder to triangulate on the Sky Pirate when they know him to be at his "home port." The "revenue service" seems to be a variation on the Revenue Marines, better known to us as the US Coast Guard.

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.

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.

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.

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.

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.

A mixed graph G = (V, E, A) is a mathematical object consisting of a set of vertices (or nodes) V, a set of (undirected) edges E, and a set of directed edges (or arcs) A.

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.

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.

The name stems from the fact that, in a moral graph, two nodes that have a common child are required to be married by sharing an edge. Moralization may also be applied to mixed graphs, called in this context "chain graphs". In a chain graph, a connected component of the undirected subgraph is called a chain. Moralization adds an undirected edge between any two vertices that both have outgoing edges to the same chain, and then forgets the orientation of the directed edges of the graph.

A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. It is unilaterally connected or unilateral (also called semiconnected) if it contains a directed path from to or a directed path from to for every pair of vertices .Chapter 11: Digraphs: Principle of duality for digraphs: Definition It is strongly connected, or simply strong, if it contains a directed path from to and a directed path from to for every pair of vertices .

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.

This problem is named after the postman and his challenge to deliver mail in any order he may choose, but minimizing his costs such as time or travel distance. It is also sometimes called the undirected chinese postman problem. The undirected rural postman problem (URPP) aims to minimize the total cost of a route that maps the entire network, or in more specific cases, a route that maps every edge that requires a service. If the whole network must be mapped, the route that maps the entire network is called a covering tour.

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.

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.

The number of vertices must be doubled because each undirected edge corresponds to two directed arcs and thus the degree of a vertex in the directed graph is twice the degree in the undirected graph. :Rahman-Kaykobad (2005). A simple graph with n vertices has a Hamiltonian path if, for every non-adjacent vertex pairs the sum of their degrees and their shortest path length is greater than n. The above theorem can only recognize the existence of a Hamiltonian path in a graph and not a Hamiltonian Cycle.

Bulatov proved a dichotomy theorem for domains of three elements. Another dichotomy theorem for constraint languages is the Hell- Nesetril theorem, which shows a dichotomy for problems on binary constraints with a single fixed symmetric relation. In terms of the homomorphism problem, every such problem is equivalent to the existence of a homomorphism from a relational structure to a given fixed undirected graph (an undirected graph can be regarded as a relational structure with a single binary symmetric relation). The Hell-Nesetril theorem proves that every such problem is either polynomial-time or NP-complete.

For any connected undirected graph G with maximum degree Δ, the chromatic number of G is at most Δ, unless G is a complete graph or an odd cycle, in which case the chromatic number is Δ + 1\.

Researchers have implicated MGTOW communities in online harassment of women. Wright et al., publishing in Information, Communication & Society in 2020, wrote that "MGTOW propagate extensive and wide-ranging passive or undirected harassment and misogyny on Twitter." Ribeiro et al.

Cycle rank was introduced by in the context of star height of regular languages. It was rediscovered by as a generalization of undirected tree-depth, which had been developed beginning in the 1980s and applied to sparse matrix computations .

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.

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).

PPA (standing for "Polynomial time Parity Argument") is the class of problems whose solution is guaranteed by the handshaking lemma: any undirected graph with an odd degree vertex must have another odd degree vertex. It contains the subclasses PWPP and PPAD.

21, 1975, pp. 89-96. and "Ajtai Fagin games" Miklos Ajtai and Ronald Fagin, "Reachability is harder for directed than for undirected finite graphs". Journal of Symbolic Logic 55, 1, March 1990, pp. 113-150. Preliminary version appeared in Proc.

Various combinatorial problems have been reduced to the Chinese Postman Problem, including finding a maximum cut in a planar graph and a minimum-mean length circuit in an undirected graph.A. Schrijver, Combinatorial Optimization, Polyhedra and Efficiency, Volume A, Springer. (2002).

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.

An example of a Markov random field. Each edge represents dependency. In this example: A depends on B and D. B depends on A and D. D depends on A, B, and E. E depends on D and C. C depends on E. In the domain of physics and probability, a Markov random field (often abbreviated as MRF), Markov network or undirected graphical model is a set of random variables having a Markov property described by an undirected graph. In other words, a random field is said to be a Markov random field if it satisfies Markov properties.

A 1-forest (a maximal pseudoforest), formed by three 1-trees In graph theory, a pseudoforest is an undirected graphThe kind of undirected graph considered here is often called a multigraph or pseudograph, to distinguish it from a simple graph. in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest.

A GraphML file consists of an XML file containing a `graph` element, within which is an unordered sequence of `node` and `edge` elements. Each `node` element should have a distinct `id` attribute, and each `edge` element has `source` and `target` attributes that identify the endpoints of an edge by having the same value as the `id` attributes of those endpoints. Here is what a simple undirected graph with two nodes and one edge between them looks like: Additional features of the GraphML language allow its users to specify whether edges are directed or undirected, and to associate additional data with vertices or edges.

In real network problems, people are interested in determining the likelihood of occurring double links (with opposite directions) between vertex pairs. This problem is fundamental for several reasons. First, in the networks that transport information or material (such as email networks, World Wide Web (WWW), World Trade Web, or Wikipedia ), mutual links facilitate the transportation process. Second, when analyzing directed networks, people often treat them as undirected ones for simplicity; therefore, the information obtained from reciprocity studies helps to estimate the error introduced when a directed network is treated as undirected (for example, when measuring the clustering coefficient).

A more general version of the theorem applies to list coloring: given any connected undirected graph with maximum degree Δ that is neither a clique nor an odd cycle, and a list of Δ colors for each vertex, it is possible to choose a color for each vertex from its list so that no two adjacent vertices have the same color. In other words, the list chromatic number of a connected undirected graph G never exceeds Δ, unless G is a clique or an odd cycle. This has been proved by . For certain graphs, even fewer than Δ colors may be needed.

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.

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..

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.

NodeXL Pro contains a library of commonly used graph metrics: centrality, clustering coefficient, diameter. NodeXL differentiates between directed and undirected networks. NodeXL Pro implements a variety of community detection algorithms to allow the user to automatically discover clusters in their social networks.

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.

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).

Most algorithms in the field of NM discovery are used to find induced sub-graphs of a network. In 2008, Noga Alon et al. introduced an approach for finding non-induced sub- graphs too. Their technique works on undirected networks such as PPI ones.

One of DeLaViña's results in graph theory is related to an inequality showing that every undirected graph has an independent set that is at least as large as its radius; DeLaViña showed that the graphs with no larger independent set always contain a Hamiltonian path.

The interior of St. Marien can be accessed via the four portals. The Gothic hall has a compact, undirected appearance. This consistent spatial impression causes misleading diagonal perspectives. The lack of a transept intensifies the feeling of closeness, as well as the effect of uniformity.

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.

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.

The recurrence matrix of a recurrence plot can be considered as the adjacency matrix of an undirected and unweighted network. This allows for the analysis of time series by network measures. Applications range from detection of regime changes over characterizing dynamics to synchronization analysis.

In 1900 the height was doubled by creating a rectangular brick building on a granite base. A large foghorn alerted ships in fog and in bad visibility. Harmaja received the world's first directed and undirected radio beacon in 1936. The lighthouse is fully automated today.

We define an undirected graph to be a set of vertices and edges such that each edge has two vertices (which may coincide) as endpoints. That is, we allow multiple edges (edges with the same pair of endpoints) and loops (edges whose two endpoints are the same vertex). A subgraph of a graph is the graph formed by any subsets of its vertices and edges such that each edge in the edge subset has both endpoints in the vertex subset. A connected component of an undirected graph is the subgraph consisting of the vertices and edges that can be reached by following edges from a single given starting vertex.

The definition and properties of Eulerian trails, cycles and graphs are valid for multigraphs as well. An Eulerian orientation of an undirected graph G is an assignment of a direction to each edge of G such that, at each vertex v, the indegree of v equals the outdegree of v. Such an orientation exists for any undirected graph in which every vertex has even degree, and may be found by constructing an Euler tour in each connected component of G and then orienting the edges according to the tour.. Every Eulerian orientation of a connected graph is a strong orientation, an orientation that makes the resulting directed graph strongly connected.

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.

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.

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.

After the death of her first husband, Goldfrank took a field trip to Alberta, Canada. Her findings led to the 1945 monograph, Changing Configurations in the Social Organization of a Blackfoot Tribe During the Reserve Period. Goldfrank published her memoirs, Notes on an Undirected Life, in 1978.

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.

In an undirected graph, a widest path may be found as the path between the two vertices in the maximum spanning tree of the graph, and a minimax path may be found as the path between the two vertices in the minimum spanning tree. In any graph, directed or undirected, there is a straightforward algorithm for finding a widest path once the weight of its minimum-weight edge is known: simply delete all smaller edges and search for any path among the remaining edges using breadth first search or depth first search. Based on this test, there also exists a linear time algorithm for finding a widest path in an undirected graph, that does not use the maximum spanning tree. The main idea of the algorithm is to apply the linear-time path-finding algorithm to the median edge weight in the graph, and then either to delete all smaller edges or contract all larger edges according to whether a path does or does not exist, and recurse in the resulting smaller graph.

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.

When he emerges from this state, he recounts three types of "contact" (phasso): # "emptiness" (suññato), # "signless" (animitto), # "undirected" (appaihito).SN 41.6. See, e.g., Thanissaro Bhikkhu (trans.) (2004), "SN 41.6 Kamabhu Sutta: With Kamabhu (On the Cessation of Perception & Feeling)," retrieved Feb 4 2009 from "Access to Insight" at www.accesstoinsight.

Big Sur is also the location of a Catholic monastery, the New Camaldoli Hermitage. The Hermitage in Big Sur was founded in 1957. It rents a few simple rooms for visitors who would like to engage in silent meditation and contemplation. Normally all retreats are silent and undirected.

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.

A squaregraph. In graph theory, a branch of mathematics, a squaregraph is a type of undirected graph that can be drawn in the plane in such a way that every bounded face is a quadrilateral and every vertex with three or fewer neighbors is incident to an unbounded face.

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.

A directed graph is strongly connected if and only if it has an ear decomposition, a partition of the edges into a sequence of directed paths and cycles such that the first subgraph in the sequence is a cycle, and each subsequent subgraph is either a cycle sharing one vertex with previous subgraphs, or a path sharing its two endpoints with previous subgraphs. According to Robbins' theorem, an undirected graph may be oriented in such a way that it becomes strongly connected, if and only if it is 2-edge-connected. One way to prove this result is to find an ear decomposition of the underlying undirected graph and then orient each ear consistently..

A strong orientation of a given bridgeless undirected graph may be found in linear time by performing a depth first search of the graph, orienting all edges in the depth first search tree away from the tree root, and orienting all the remaining edges (which must necessarily connect an ancestor and a descendant in the depth first search tree) from the descendant to the ancestor.See e.g. and . If an undirected graph with bridges is given, together with a list of ordered pairs of vertices that must be connected by directed paths, it is possible in polynomial time to find an orientation of that connects all the given pairs, if such an orientation exists.

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.

For, if the vertex separation number of a topological ordering is at most w, the minimum vertex separation among all orderings can be no larger, so the undirected graph formed by ignoring the orientations of the DAG described above must have pathwith at most w. It is possible to test whether this is the case, using the known fixed-parameter- tractable algorithms for pathwidth, and if so to find a path-decomposition for the undirected graph, in linear time given the assumption that w is a constant. Once a path decomposition has been found, a topological ordering of width w (if one exists) can be found using dynamic programming, again in linear time.

The MaxCliqueDyn algorithm is an algorithm for finding a maximum clique in an undirected graph. It is based on a basic algorithm (MaxClique algorithm) which finds a maximum clique of bounded size. The bound is found using improved coloring algorithm. The MaxCliqueDyn extends MaxClique algorithm to include dynamically varying bounds.

Middendorf-Ziv (MZ) proposed a growing directed graph modeling biological network dynamics. A prototype is chosen at random and duplicated. The prototype or progenitor node has edges pruned with probability β and edges added with probability α<<β. Based loosely on the undirected protein network model of Sole et al.

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.

Neil follows Madeleine to an escarpment, where Murder commands his zombie guardians to kill Neil. Bruner approaches Murder and knocks him out, breaking Murder's mental control over his zombies. Undirected, the zombies topple off the cliff. Murder awakens and eludes Neil and Bruner, but Charles pushes Murder off the cliff.

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.

Similarly, binary search trees are the case where the edges to the left or right subtrees are given when the queried vertex is unequal to the target. For all undirected, positively weighted graphs, there is an algorithm that finds the target vertex in O(\log n) queries in the worst case.

Weighted Network :In reality, not all edges shares the same importance or weight (connections in a social network and keystone species in a food web, for example). A weighted network adds such element to its connections. It is widely used in genomic and systems biologic applications. Trees :Undirected networks with no closed loops.

Roberts' research concerns graph theory and combinatorics, and their applications in modeling problems in the social sciences and biology. Among his contributions to pure mathematics, he is known for introducing the concept of boxicity, the minimum dimension needed to represent a given undirected graph as an intersection graph of axis-parallel boxes..

The biclique partition problem takes as input an arbitrary undirected graph, and asks for a partition of its edges into a minimum number of complete bipartite graphs. It is NP-hard, but fixed-parameter tractable. The best approximation algorithm known for the problem has an approximation ratio of O(n/\log n).

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).

A gradient network is a directed subnetwork of an undirected "substrate" network in which each node has an associated scalar potential and one out-link that point to the node with the smallest (or largest) potential in its neighborhood, defined as the reunion of itself and its nearest neighbors on the substrate networks.

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.

It is also possible to compute a simpler invariant of directed graphs by ignoring the directions of the edges and computing the circuit rank of the underlying undirected graph. This principle forms the basis of the definition of cyclomatic complexity, a software metric for estimating how complicated a piece of computer code is.

In a hypergraph, an edge can join more than two vertices. An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices. Every graph gives rise to a matroid.

The arboricity of an undirected graph is the minimum number of forests into which its edges can be partitioned. Equivalently it is the minimum number of spanning forests needed to cover all the edges of the graph. The Nash-Williams theorem provides necessary and sufficient conditions for when a graph is k-arboric.

The cube-connected cycles of order 3, arranged geometrically on the vertices of a truncated cube. In graph theory, the cube-connected cycles is an undirected cubic graph, formed by replacing each vertex of a hypercube graph by a cycle. It was introduced by for use as a network topology in parallel computing.

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.

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.

In social network analysis, the co-stardom network represents the collaboration graph of film actors i.e. movie stars. The co-stardom network can be represented by an undirected graph of nodes and links. Nodes correspond to the movie star actors and two nodes are linked if they co-starred (performed) in the same movie.

Peabody even persuaded the fire chief to wire a fire alarm bell to his house. Polly's life was difficult during the war years, and when her husband returned home and resumed drinking, her commitment to her marriage was further weakened. Polly felt that her husband was a well-educated but undirected man and a reluctant father.

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.

GraphML is an XML-based file format for graphs. The GraphML file format results from the joint effort of the graph drawing community to define a common format for exchanging graph structure data. It uses an XML-based syntax and supports the entire range of possible graph structure constellations including directed, undirected, mixed graphs, hypergraphs, and application- specific attributes..

The central nave is only slightly wider than the side aisles. This results in merely a weak accentuation of the longitudinal axis of the church. The bays of the side aisles counteract this slight longitudinal alignment by opening up their broadsides to the central nave, thus emphasising the lateral alignment. This gives the impression of undirected space.

Fig.5 Association An association is a structural relationship that specifies how concepts are connected to another. It can connect two concepts (binary association) or more than two concepts (n-ary association). An association is represented with an undirected solid line. To give a meaning to the association, a name and name direction can be provided.

In the mathematical area of graph theory, an undirected graph G is strongly chordal if it is a chordal graph and every cycle of even length (≥ 6) in G has an odd chord, i.e., an edge that connects two vertices that are an odd distance (>1) apart from each other in the cycle., Definition 3.4.1, p. 43.

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.

The global clustering coefficient is based on triplets of nodes. A triplet is three nodes that are connected by either two (open triplet) or three (closed triplet) undirected ties. A triangle graph therefore includes three closed triplets, one centered on each of the nodes (n.b. this means the three triplets in a triangle come from overlapping selections of nodes).

There are five different metrics that can be used to calculate distinctiveness centrality – namely D1, D2, D3, D4 and D5. They only differ with regard to the weighting factor used. In addition, only D1, D3 and D4 are designed to consider arc weights. Formulas are presented for a (weighted) undirected graph G, made of n nodes and m arcs.

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.

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.

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, .

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.

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.

Mantel's theorem states that any n-vertex undirected graph with at least n2/4 edges, and no multiple edges or self-loops, either contains a triangle or it is the complete bipartite graph Kn/2,n/2. This theorem can be strengthened: any undirected Hamiltonian graph with at least n2/4 edges is either pancyclic or Kn/2,n/2. There exist n-vertex Hamiltonian directed graphs with n(n + 1)/2 − 3 edges that are not pancyclic, but every Hamiltonian directed graph with at least n(n + 1)/2 − 1 edges is pancyclic. Additionally, every n-vertex strongly connected directed graph in which each vertex has degree at least n (counting incoming and outgoing edges together) is either pancyclic or it is a complete bipartite directed graph..

What route should I take to minimize the total distance?” Figure 14 shows an undirected graph for this roadmap, the numbers indicating the shortest distances between neighboring capital cities. The problem is to choose a subset of these edges that form a Hamiltonian path of smallest total length. Every Hamiltonian path in this graph must either start or end at Augusta, Maine(ME).

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.

The physical network topology can be directly represented in a network diagram, as it is simply the physical graph represented by the diagrams, with network nodes as vertices and connections as undirected or direct edges (depending on the type of connection). The logical network topology can be inferred from the network diagram if details of the network protocols in use are also given.

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.

An example of how Chinese Whispers works in action. The different colors represent different classes. The algorithm works in the following way in an undirected unweighted graph: # All nodes are assigned to a distinct class (The number of initial classes equals the number of nodes). # Then all of the network nodes are selected one by one in a random order.

Alien Discussions: Proceedings of the Abduction Study Conference. Cambridge: North Cambridge Press, 1994. pp. 83–85. These feelings manifest as a compulsive desire to be at a certain place at a certain time or as expectations that something "familiar yet unknown," will soon occur. Abductees also report feeling severe, undirected anxiety at this point even though nothing unusual has actually occurred yet.

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.

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.

Graph of an example equivalence with 7 classes An undirected graph may be associated to any symmetric relation on a set , where the vertices are the elements of , and two vertices and are joined if and only if . Among these graphs are the graphs of equivalence relations; they are characterized as the graphs such that the connected components are cliques.

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.

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.

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.

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\.

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.

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.

An edge contraction is an operation which removes an edge from a graph while simultaneously merging the two vertices it used to connect. An undirected graph H is a minor of another undirected graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, and deleting some isolated vertices. The order in which a sequence of such contractions and deletions is performed on G does not affect the resulting graph H. Graph minors are often studied in the more general context of matroid minors. In this context, it is common to assume that all graphs are connected, with self-loops and multiple edges allowed (that is, they are multigraphs rather than simple graphs); the contraction of a loop and the deletion of a cut-edge are forbidden operations.

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.

Where organs can be purchased, the supply increases. Healthy humans have two kidneys, a redundancy that enables living donors (inter vivos) to give a kidney to someone who needs it. The most common transplants are to close relatives, but people have given kidneys to other friends. The rarest type of donation is the undirected donation whereby a donor gives a kidney to a stranger.

For unidimensional networks, the HITS algorithm has been originally introduced by Jon Kleinberg to rate Web Pages. The basic assumption of the algorithm is that relevant pages, named authorities, are pointed by special Web pages, named hubs. This mechanism can be mathematically described by two coupled equations which reduce to two eigenvalue problems. When the network is undirected, Authority and Hub centrality are equivalent to eigenvector centrality.

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.

Graph500 is the first benchmark for data-intensive supercomputing problems. This benchmark generates an edge tuple with two endpoints at first. Then the kernel 1 will constructs an undirected graph, in which weight of edge will not be assigned if only kernel 2 runs afterwards. Users can choose to run BFS in kernel 2 and/or Single-Source-Shortest-Path in kernel 3 on the constructed graph.

Symmetric space-bounded computation. Theoretical Computer Science. pp.161-187. 1982. who were looking for a class in which to place USTCON, the problem asking whether there is a path between two given vertices s,t in an undirected graph. Until this time, it could be placed only in NL, despite seeming not to require nondeterminism (the asymmetric variant STCON was known to be complete for NL).

Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.

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.

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.

A polytree, a directed acyclic graph formed by assigning an orientation to each edge of an undirected tree, may be viewed as a special case of a multitree. The set of all vertices connected to any vertex in a multitree forms an arborescence. The word "multitree" has also been used to refer to a series-parallel partial order,. or to other structures formed by combining multiple trees.

A red–black tree plotted by Graphviz. Undirected graph showing adjacency of the 48 contiguous United States Graphviz (short for Graph Visualization Software) is a package of open-source tools initiated by AT&T; Labs Research for drawing graphs specified in DOT language scripts. It also provides libraries for software applications to use the tools. Graphviz is free software licensed under the Eclipse Public License.

The node definitions are separated from the edge definitions by a line containing the "#" character. Each edge definition is another line of text, starting with the two IDs for the endpoints of the edge separated by a space. If the edge has a label, it appears on the same line after the endpoint IDs. The graph may be interpreted as a directed or undirected graph.

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.

The links in the network will be undirected if the proximities are symmetrical for every pair of entities. Symmetrical proximities mean that the order of the entities is not important, so the proximity of i and j is the same as the proximity of j and i for all pairs i,j. If the proximities are not symmetrical for every pair, the links will be directed.

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.

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; .

41, 46. Postman defines technopoly as a "totalitarian technocracy", which demands the "submission of all forms of cultural life to the sovereignty of technique and technology".Postman (1993), p. 52 Echoing Ellul's 1964 conceptualisation of technology as autonomous, "self-determinative" independently of human action, and undirected in its growth,Tiles & Oberdiek (1995), p. 22. technology in a time of Technopoly actively eliminates all other ‘thought-worlds’.

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.

Vartanov's debut film, The Color of Armenian Land, marked the beginning of his trademark style, afterwards dubbed as the "direction of undirected action." This documentary, a silent commentary on the technique of painter Martiros Saryan, also featured Vartanov's friends, the dissident artists Minas Avetisyan and Sergei Parajanov. Due to this the film was censored and suppressed; leading up to Avetisyan's assassination and Parajanov's imprisonment shortly after.

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.

That would account for Fido's inability to find a record of his death in England and Wales during the probable period of his life.Kendell, p. 80 Nigel Cawthorne dismissed Cohen as a likely suspect because in the asylum his assaults were undirected, and his behaviour was wild and uncontrolled, whereas the Ripper seemed to attack specifically and quietly.Cawthorne, Nigel (2000) "Foreword", in Knight, p.

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.

That is, the distortion of graphs in the family is bounded by a constant that depends on the family but not on the individual graphs. For instance, the planar graphs are closed under minors. Therefore, it would follow from the GNRS conjecture that the planar graphs have bounded distortion. An alternative formulation involves analogues of the max-flow min-cut theorem for undirected multi-commodity flow problems.

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.

Two non-isomorphic graphs with the same degree sequence (3, 2, 2, 2, 2, 1, 1, 1). The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees;Diestel p.216 for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant so isomorphic graphs have the same degree sequence.

ObsTCP is a low cost protocol intended to protect TCP traffic, without requiring public key certificates, the services of Certificate Authorities, or a complex Public Key Infrastructure. It is intended to suppress the use of undirected surveillance to trawl unencrypted traffic, rather than protect against man in the middle attack. The software presently supports the Salsa20/8 stream cipher and Curve25519 elliptic-curve Diffie Hellman function.

More generally, any edge-weighted undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components. There are quite a few use cases for minimum spanning trees. One example would be a telecommunications company trying to lay cable in a new neighborhood. If it is constrained to bury the cable only along certain paths (e.g.

If e is in the first subset of edges at v, these two edges are from u0 into v0 and from v1 into u1, while if e is in the second subset, the edges are from u0 into v1 and from v0 into u1. In the other direction, given a skew- symmetric graph G, one may form a polar graph that has one vertex for every corresponding pair of vertices in G and one undirected edge for every corresponding pair of edges in G. The undirected edges at each vertex of the polar graph may be partitioned into two subsets according to which vertex of the polar graph they go out of and come into. A regular path or cycle of a skew-symmetric graph corresponds to a path or cycle in the polar graph that uses at most one edge from each subset of edges at each of its vertices.

Any partial order may be represented (usually in more than one way) by a directed acyclic graph in which there is a path from x to y whenever x and y are elements of the partial order with . The graphs that represent series-parallel partial orders in this way have been called vertex series parallel graphs, and their transitive reductions (the graphs of the covering relations of the partial order) are called minimal vertex series parallel graphs. Directed trees and (two- terminal) series parallel graphs are examples of minimal vertex series parallel graphs; therefore, series parallel partial orders may be used to represent reachability relations in directed trees and series parallel graphs. The comparability graph of a partial order is the undirected graph with a vertex for each element and an undirected edge for each pair of distinct elements x, y with either or .

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.

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.

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.

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.

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.

In graph theoretic terms, the states are Eulerian orientations of an underlying 4-regular undirected graph. The partition function also counts the number of nowhere-zero 3-flows. For two-dimensional models, the lattice is taken to be the square lattice. For more realistic models, one can use a three-dimensional lattice appropriate to the material being considered; for example, the hexagonal ice lattice is used to analyse ice.

For instance, the condition that a graph does not have any isolated vertices may be expressed by the sentence :\forall u\exists v(u\sim v) where the \sim symbol indicates the undirected adjacency relation between two vertices. This sentence can be interpreted as meaning that for every vertex u there is another vertex v that is adjacent to u., Section 1.2, "What Is a First Order Theory?", pp. 15–17.

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.

For certain undirected graphical models, it is possible to efficiently perform exact inference via message passing, or belief propagation algorithms. These algorithms follow a simple iterative pattern: each variable passes its "beliefs" about its neighbors' marginal distributions, then uses the incoming messages about its own value to update its beliefs. Convergence to the true marginals is guaranteed for tree-structured MRFs, but is not guaranteed for MRFs with cycles.

Bobbitt realized that there were too many activities (for example related to citizenship, health, spare time, parentship, work related activities and languages) to fit in any curriculum. A part of those activities were well taught by socialization: the so-called undirected experiences. This is why the curriculum has to aim at the particular subjects that are not sufficiency learned as a result of normal socialization, these subjects were described as shortcomings.

Using the LLL-algorithm, Frank, and his student, Éva Tardos developed a general method, which could transform some polynomial-time algorithms into strongly polynomial.. He solved the problem of finding the minimum number of edges to be added to a given undirected graph so that in the resulting graph the edge-connectivity between any two vertices u and v is at least a predetermined number f(u,v)..

The running time of this algorithm is O(EV). A faster implementation of the algorithm due to Robert Tarjan runs in time O(E \log V) for sparse graphs and O(V^2) for dense graphs. This is as fast as Prim's algorithm for an undirected minimum spanning tree. In 1986, Gabow, Galil, Spencer, Compton, and Tarjan produced a faster implementation, with running time O(E + V \log V).

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.

Let the adjacency matrix for the network be represented by A, where A_=0 means there's no edge (no interaction) between nodes v and w and A_{vw} = 1 means there is an edge between the two. Also for simplicity we consider an undirected network. Thus A_{vw} = A_{wv}. (It is important to note that multiple edges may exist between two nodes, but here we assess the simplest case).

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.

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.

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.

The Russian Revolution of 1905 was a wave of mass political unrest through vast areas of the Russian Empire. Some of it was directed against the government, while some was undirected. It included terrorism, worker strikes, peasant unrests, and military mutinies. It led to the establishment of the limited constitutional monarchy,Russian Constitution of 1906 the establishment of State Duma of the Russian Empire, and the multi-party system.

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.

Symmetric TSP with four cities TSP can be modelled as an undirected weighted graph, such that cities are the graph's vertices, paths are the graph's edges, and a path's distance is the edge's weight. It is a minimization problem starting and finishing at a specified vertex after having visited each other vertex exactly once. Often, the model is a complete graph (i.e., each pair of vertices is connected by an edge).

But she is a griffin, appearing at first glance to be frightening and inscrutable, and her father does not approve of university education. The classmates quickly become fast friends, and two soon fall in love. They form common opinions of their teachers and courses, and undertake together some undirected extracurricular study. They all run afoul of Wizard Wermacht, a domineering man who teaches multiple subjects in a routine fashion.

Define the degree of saturation of a vertex as the number of different colours in its neighbourhood. Given a simple, undirected graph G compromising vertex set V and edge set E: # Generate a degree ordering of V. # Select a vertex of maximal degree and colour it with the first colour. # Consider a vertex with the highest degree of saturation. Break ties by considering that vertex with the highest degree.

Finding the optimal order in which to eliminate variables is an NP-hard problem. As such there are heuristics one may follow to better optimize performance by order: # Minimum Degree: Eliminate the variable which results in constructing the smallest factor possible. # Minimum Fill: By constructing an undirected graph showing variable relations expressed by all CPTs, eliminate the variable which would result in the least edges to be added post elimination.

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.

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.

An example of an ear decomposition of a graph containing 3 ears. In graph theory, an ear of an undirected graph G is a path P where the two endpoints of the path may coincide, but where otherwise no repetition of edges or vertices is allowed, so every internal vertex of P has degree two in G. An ear decomposition of an undirected graph G is a partition of its set of edges into a sequence of ears, such that the one or two endpoints of each ear belong to earlier ears in the sequence and such that the internal vertices of each ear do not belong to any earlier ear. Additionally, in most cases the first ear in the sequence must be a cycle. An open ear decomposition or a proper ear decomposition is an ear decomposition in which the two endpoints of each ear after the first are distinct from each other.

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.

The clique complex of a graph. Cliques of size one are shown as small red disks; cliques of size two are shown as black line segments; cliques of size three are shown as light blue triangles; and cliques of size four are shown as dark blue tetrahedra. Clique complexes, flag complexes, and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques (complete subgraphs) of an undirected graph. The clique complex X(G) of an undirected graph 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 cliques of G. Any subset of a clique is itself a clique, so this family of sets meets the requirement of an abstract simplicial complex that every subset of a set in the family should also be in the family.

Roughly speaking, a Feynman diagram is called connected if all vertices and propagator lines are linked by a sequence of vertices and propagators of the diagram itself. If one views it as an undirected graph it is connected. The remarkable relevance of such diagrams in QFTs is due to the fact that they are sufficient to determine the quantum partition function . More precisely, connected Feynman diagrams determine :i W[J]\equiv \ln Z[J].

Mixed graphs are also used as graphical models for Bayesian inference. In this context, an acyclic mixed graph (one with no cycles of directed edges) is also called a chain graph. The directed edges of these graphs are used to indicate a causal connection between two events, in which the outcome of the first event influences the probability of the second event. Undirected edges, instead, indicate a non-causal correlation between two events.

In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree subgraphs. This set of subgraphs can be described algebraically as a vector space over the two- element finite field. The dimension of this space is the circuit rank of the graph. The same space can also be described in terms from algebraic topology as the first homology group of the graph.

In 1932, the young Henri Cartier-Bresson, at the time an undirected photographer who catalogued his travels and his friends, saw the Munkácsi photograph Three Boys at Lake Tanganyika, taken on a beach in Liberia. Cartier-Bresson later said, > For me this photograph was the spark that ignited my enthusiasm. I suddenly > realized that, by capturing the moment, photography was able to achieve > eternity. It is the only photograph to have influenced me.

The technique, which culminates in Lemma 7.10 on p.218 of Imrich and Klavžar, consists of applying an algorithm of to list all 4-cycles in the graph G, forming an undirected graph having as its vertices the edges of G and having as its edges the opposite sides of a 4-cycle, and using the connected components of this derived graph to form hypercube coordinates. An equivalent algorithm is , Algorithm H, p. 69.

"The Chart", an undirected labeled graph in which nodes represent individuals and lines represent affairs or hookups, is a recurring plot element throughout the series.Elizabeth Jensen, "‘The L Word’ Spins Off Its Chart", NY Times, 2006-12-18. Originally, The L Word was to be based around a gay woman Kit Porter, and "The Chart" was tattooed on her back. In season 4, Alice launches The Chart as a social networking service.

For an undirected graph, the degree of a vertex is equal to the number of adjacent vertices. A special case is a loop, which adds two to the degree. This can be understood by letting each connection of the loop edge count as its own adjacent vertex. In other words, a vertex with a loop "sees" itself as an adjacent vertex from both ends of the edge thus adding two, not one, to the degree.

Moravec explains, "Their initial promises to DARPA had been much too optimistic. Of course, what they delivered stopped considerably short of that. But they felt they couldn't in their next proposal promise less than in the first one, so they promised more." However, there was another issue: since the passage of the Mansfield Amendment in 1969, DARPA had been under increasing pressure to fund "mission- oriented direct research, rather than basic undirected research".

In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex s can reach a vertex t (and t is reachable from s) if there exists a sequence of adjacent vertices (i.e. a path) which starts with s and ends with t. In an undirected graph, reachability between all pairs of vertices can be determined by identifying the connected components of the graph.

Before being fully conceptualized by Scott, British historian E.P. Thompson was the first to use the term "moral economy" in Moral Economy of the English Crowd in the Eighteenth Century. In this work, he discussed English bread riots, regular, localized form of rebellion by English peasants all through the 18th century. Such events, Thompson argues, have been routinely dismissed as "riotous", with the connotation of being disorganized, spontaneous, undirected, and undisciplined. In other words, anecdotal.

Restricted Boltzmann machines (RBMs) are often used as a building block for multilayer learning architectures. An RBM can be represented by an undirected bipartite graph consisting of a group of binary hidden variables, a group of visible variables, and edges connecting the hidden and visible nodes. It is a special case of the more general Boltzmann machines with the constraint of no intra-node connections. Each edge in an RBM is associated with a weight.

Another characterization is possible for graphs with a single source. In this case an upward planar embedding must have the source on the outer face, and every undirected cycle of the graph must have at least one vertex at which both cycle edges are incoming (for instance, the vertex with the highest placement in the drawing). Conversely, if an embedding has both of these properties, then it is equivalent to an upward embedding., p.

It makes the claim that "certain features of the universe and of living things are best explained by an intelligent cause, not an undirected process such as natural selection." It has been viewed as a "scientific" approach to creationism by creationists, but is widely rejected as pseudoscience by the science community—primarily because intelligent design cannot be tested and rejected like scientific hypotheses (see for example, List of scientific bodies explicitly rejecting intelligent design).

Minimal Bottleneck Spanning Tree In an undirected graph and a function , let be the set of all spanning trees Ti. Let B(Ti) be the maximum weight edge for any spanning tree Ti. We define subset of minimum bottleneck spanning trees S′ such that for every and we have for all i and k. The graph on the right is an example of MBST, the red edges in the graph form a MBST of .

This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs structure. For M ≃ pN, where N is the maximal number of edges possible, the two most widely used models, G(n,M) and G(n,p), are almost interchangeable.Bollobas, B. and Riordan, O.M. "Mathematical results on scale-free random graphs" in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed.

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.

The seven graphs in the Petersen family. No matter how these graphs are embedded into three-dimensional space, some two cycles will have nonzero linking number. In two dimensions, only the planar graphs may be embedded into the Euclidean plane without crossings, but in three dimensions, any undirected graph may be embedded into space without crossings. However, a spatial analogue of the planar graphs is provided by the graphs with linkless embeddings and knotless embeddings.

There is no one-to-one correspondence between such trees and trees as data structure. We can take an arbitrary undirected tree, arbitrarily pick one of its vertices as the , make all its edges directed by making them point away from the root node – producing an arborescence – and assign an order to all the nodes. The result corresponds to a tree data structure. Picking a different root or different ordering produces a different one.

In the symmetric TSP, the distance between two cities is the same in each opposite direction, forming an undirected graph. This symmetry halves the number of possible solutions. In the asymmetric TSP, paths may not exist in both directions or the distances might be different, forming a directed graph. Traffic collisions, one-way streets, and airfares for cities with different departure and arrival fees are examples of how this symmetry could break down.

A log-linear model is graphical if, whenever the model contains all two-factor terms generated by a higher-order interaction, the model also contains the higher-order interaction. As a direct-consequence, graphical models are hierarchical. Moreover, being completely determined by its two-factor terms, a graphical model can be represented by an undirected graph, where the vertices represent the variables and the edges represent the two-factor terms included in the model.

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.

Pure undirected research of the kind that had gone on in the 1960s would no longer be funded by DARPA. Researchers now had to show that their work would soon produce some useful military technology. AI research proposals were held to a very high standard. The situation was not helped when the Lighthill report and DARPA's own study (the American Study Group) suggested that most AI research was unlikely to produce anything truly useful in the foreseeable future.

The Rado graph: for instance there is an edge from 0 to 3 because the 0th bit of 3 is non zero. Ackermann in 1937 and Richard Rado in 1964 used this predicate to construct the infinite Rado graph. In their construction, the vertices of this graph correspond to the non-negative integers, written in binary, and there is an undirected edge from vertex i to vertex j, for i < j, when BIT(j,i) is nonzero..

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.

The monochromatic triangle problem takes as input an n-node undirected graph G(V,E) with node set V and edge set E. The output is a Boolean value, true if the edge set E of G can be partitioned into two disjoint sets E1 and E2, such that both of the two subgraphs G1(V,E1) and G2(V,E2) are triangle-free graphs, and false otherwise. This decision problem is NP-complete.. A1.1: GT6, pg.191.

A., 217, 295–305) or the torus (the bivariate von Mises distribution). The matrix von Mises–Fisher distribution is a distribution on the Stiefel manifold, and can be used to construct probability distributions over rotation matrices. The Bingham distribution is a distribution over axes in N dimensions, or equivalently, over points on the (N − 1)-dimensional sphere with the antipodes identified. For example, if N = 2, the axes are undirected lines through the origin in the plane.

Any pair of vertices in such a graph can reach each other if and only if they belong to the same connected component; therefore, in such a graph, reachability is symmetric (s reaches t iff t reaches s). The connected components of an undirected graph can be identified in linear time. The remainder of this article focuses on the more difficult problem of determining pairwise reachability in a directed graph (which, incidentally, need not be symmetric).

The BEST theorem shows that the number of Eulerian circuits in directed graphs can be computed in polynomial time, a problem which is #P-complete for undirected graphs.Brightwell and Winkler, "Note on Counting Eulerian Circuits", CDAM Research Report LSE-CDAM-2004-12, 2004. It is also used in the asymptotic enumeration of Eulerian circuits of complete and complete bipartite graphs.Brendan McKay and Robert W. Robinson, Asymptotic enumeration of eulerian circuits in the complete graph, Combinatorica, 10 (1995), no.

In the random surfing model, webgraphs are presented as a sequence of directed graphs G_t,t = 1,2,\ldots such that a graph G_t has t vertices and t edges. The process of defining graphs is parameterized with a probability p, thus we let q= 1-p. Nodes of the model arrive one at time, forming k connections to the existing graph G_t . In some models, connections represent directed edges, and in others, connections represent undirected edges.

A free tree or unrooted tree is a connected undirected graph with no cycles. The vertices with one neighbor are the leaves of the tree, and the remaining vertices are the internal nodes of the tree. The degree of a vertex is its number of neighbors; in a tree with more than one node, the leaves are the vertices of degree one. An unrooted binary tree is a free tree in which all internal nodes have degree exactly three.

The conjecture was disproved by Aanderaa, who exhibited a directed graph property (the property of containing a "sink") which required only O(n) queries to test. A sink, in a directed graph, is a vertex of indegree n-1 and outdegree 0. This property can be tested with less than 3n queries . An undirected graph property which can also be tested with O(n) queries is the property of being a scorpion graph, first described in .

The key difference from the more recent Barabási–Albert model is that the Price model produces a graph with directed edges while the Barabási–Albert model is the same model but with undirected edges. The direction is central to the citation network application which motivated Price. This means that the Price model produces a directed acyclic graph and these networks have distinctive properties. For example, in a directed acyclic graph both longest paths and shortest paths are well defined.

A demo for Union-Find when using Kruskal's algorithm to find minimum spanning tree. Disjoint-set data structures model the partitioning of a set, for example to keep track of the connected components of an undirected graph. This model can then be used to determine whether two vertices belong to the same component, or whether adding an edge between them would result in a cycle. The Union–Find algorithm is used in high-performance implementations of unification.

The circle-based 1.1-skeleton (heavy dark edges) and 0.9-skeleton (light dashed blue edges) of a set of 100 random points in a square. In computational geometry and geometric graph theory, a β-skeleton or beta skeleton is an undirected graph defined from a set of points in the Euclidean plane. Two points p and q are connected by an edge whenever all the angles prq are sharper than a threshold determined from the numerical parameter β.

Using min heap priority queue in Prim's algorithm to find the minimum spanning tree of a connected and undirected graph, one can achieve a good running time. This min heap priority queue uses the min heap data structure which supports operations such as insert, minimum, extract-min, decrease-key. In this implementation, the weight of the edges is used to decide the priority of the vertices. Lower the weight, higher the priority and higher the weight, lower the priority.

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).

A Halin graph is a graph formed from an undirected plane tree (with no degree- two nodes) by connecting its leaves into a cycle, in the order given by the plane embedding of the tree. Equivalently, it is a polyhedral graph in which one face is adjacent to all the others. Every Halin graph is planar. Like outerplanar graphs, Halin graphs have low treewidth, making many algorithmic problems on them more easily solved than in unrestricted planar graphs..

A variant of the previously described discriminative model is the linear-chain conditional random field. This uses an undirected graphical model (aka Markov random field) rather than the directed graphical models of MEMM's and similar models. The advantage of this type of model is that it does not suffer from the so-called label bias problem of MEMM's, and thus may make more accurate predictions. The disadvantage is that training can be slower than for MEMM's.

Intuitively, an expander graph is a finite, undirected multigraph in which every subset of the vertices that is not "too large" has a "large" boundary. Different formalisations of these notions give rise to different notions of expanders: edge expanders, vertex expanders, and spectral expanders, as defined below. A disconnected graph is not an expander, since the boundary of a connected component is empty. Every connected graph is an expander; however, different connected graphs have different expansion parameters.

Interval class . In musical set theory, an interval class (often abbreviated: ic), also known as unordered pitch-class interval, interval distance, undirected interval, or "(even completely incorrectly) as 'interval mod 6'" (; ), is the shortest distance in pitch class space between two unordered pitch classes. For example, the interval class between pitch classes 4 and 9 is 5 because 9 − 4 = 5 is less than 4 − 9 = −5 ≡ 7 (mod 12). See modular arithmetic for more on modulo 12.

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 .

An Apollonian network The Goldner–Harary graph, a non-Hamiltonian Apollonian network In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.

According to Pappé, plan Dalet was the master plan for the expulsion of the Palestinians.. However, according to Gelber, Plan Dalet instructions were: In case of resistance, the population of conquered villages was to be expelled outside the borders of the Jewish state. If no resistance was met, the residents could stay put, under military rule. Palestinian belligerency in these first few months was "disorganised, sporadic and localised and for months remained chaotic and uncoordinated, if not undirected".

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.

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.

An undirected graph is a mathematical object consisting of a set of vertices and a set of edges that link pairs of vertices. The two vertices associated with each edge are called its endpoints. The graph is finite when its vertices and edges form finite sets, and infinite otherwise. A graph coloring associates each vertex with a color drawn from a set of colors, in such a way that every edge has two different colors at its endpoints.

In graph theory, a friendly-index set is a finite set of integers associated with a given undirected graph and generated by a type of graph labeling called a friendly labeling. A friendly labeling of an -vertex undirected graph is defined to be an assignment of the values 0 and 1 to the vertices of with the property that the number of vertices labeled 0 is as close as possible to the number of vertices labeled 1: they should either be equal (for graphs with an even number of vertices) or differ by one (for graphs with an odd number of vertices). Given a friendly labeling of the vertices of , one may also label the edges: a given edge is labeled with a 0 if its endpoints and have equal labels, and it is labeled with a 1 if its endpoints have different labels. The friendly index of the labeling is the absolute value of the difference between the number of edges labeled 0 and the number of edges labeled 1.

The problem of planning a freight delivery system may be modeled by a network in which the vertices represent cities and the (undirected) edges represent potential freight delivery routes between pairs of cities. Each route can achieve a certain profit, but can only be used if freight depots are constructed at both its ends, with a certain cost. The problem of designing a network that maximizes the difference between the profits and the costs can be solved as a closure problem, by subdividing each undirected edge into two directed edges, both directed outwards from the subdivision point. The weight of each subdivision point is a positive number, the profit of the corresponding route, and the weight of each original graph vertex is a negative number, the cost of building a depot in that city.. Together with open pit mining, this was one of the original motivating applications for studying the closure problem; it was originally studied in 1970, in two independent papers published in the same issue of the same journal by J. M. W. Rhys and Michel Balinski...

The original paper analysed the complexity of the problem and reported it to be PSPACE-complete. It was also shown that finding an optimal path in the case where each edge has an associated probability of being in the graph (i-SSPPR) is a PSPACE-easy but ♯P-hard problem.Papadimitriou and Yannakakis, 1989, p. 148 It was an open problem to bridge this gap, but since then both the directed and undirected versions were shown to be PSPACE-hard.

If points are in general position, the degree is at most 5. #The NNG (treated as an undirected graph with multiple nearest neighbors allowed) of a set of points in the plane or any higher dimension is a subgraph of the Delaunay triangulation, the Gabriel graph, and the Semi-Yao graph. If the points are in general position or if the single nearest neighbor condition is imposed, the NNG is a forest, a subgraph of the Euclidean minimum spanning tree.

Mixed graphs may be used to model job shop scheduling problems in which a collection of tasks is to be performed, subject to certain timing constraints. In this sort of problem, undirected edges may be used to model a constraint that two tasks are incompatible (they cannot be performed simultaneously). Directed edges may be used to model precedence constraints, in which one task must be performed before another. A graph defined in this way from a scheduling problem is called a disjunctive graph.

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.

The global clustering coefficient is the number of closed triplets (or 3 x triangles) over the total number of triplets (both open and closed). The first attempt to measure it was made by Luce and Perry (1949). This measure gives an indication of the clustering in the whole network (global), and can be applied to both undirected and directed networks (often called transitivity, see Wasserman and Faust, 1994, page 243Stanley Wasserman, Katherine Faust, 1994. Social Network Analysis: Methods and Applications.

Graph partitioning methods are an effective tools for image segmentation since they model the impact of pixel neighborhoods on a given cluster of pixels or pixel, under the assumption of homogeneity in images. In these methods, the image is modeled as a weighted, undirected graph. Usually a pixel or a group of pixels are associated with nodes and edge weights define the (dis)similarity between the neighborhood pixels. The graph (image) is then partitioned according to a criterion designed to model "good" clusters.

Emphasizing free association and undirected coincidence between music and motif, Kilbey declined to define their meanings. Sonically, the music had numerous layers, courtesy of numerous guitar overdubs and MacKillop's rich production. The interplay between Koppes and Willson-Piper dominated throughout, especially on tracks such as "Ripple," "Kings," and the epic, aptly titled "Chaos", whose lyrics were a reflection of Steve Kilbey's unsettled lifestyle at the time. Upon its release on 10 March 1992, Priest=Aura was given a mixed reception.

An Eulerian trail,Some people reserve the terms path and cycle to mean non-self- intersecting path and cycle. A (potentially) self-intersecting path is known as a trail or an open walk; and a (potentially) self-intersecting cycle, a circuit or a closed walk. This ambiguity can be avoided by using the terms Eulerian trail and Eulerian circuit when self-intersection is allowed. or Euler walk in an undirected graph is a walk that uses each edge exactly once.

"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.

The Frucht graph, a 3-regular graph whose automorphism group realizes the trivial group. Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite undirected graph. More strongly, for any finite group G there exist infinitely many non-isomorphic simple connected graphs such that the automorphism group of each of them is isomorphic to G.

In mathematics, a minimum bottleneck spanning tree (MBST) in an undirected graph is a spanning tree in which the most expensive edge is as cheap as possible. A bottleneck edge is the highest weighted edge in a spanning tree. A spanning tree is a minimum bottleneck spanning tree if the graph does not contain a spanning tree with a smaller bottleneck edge weight.Everything about Bottleneck Spanning Tree For a directed graph, a similar problem is known as Minimum Bottleneck Spanning Arborescence (MBSA).

Camerini proposed an algorithm used to obtain a minimum bottleneck spanning tree (MBST) in a given undirected, connected, edge-weighted graph in 1978. It half divides edges into two sets. The weights of edges in one set are no more than that in the other. If a spanning tree exists in subgraph composed solely with edges in smaller edges set, it then computes a MBST in the subgraph, a MBST of the subgraph is exactly a MBST of the original graph.

Links in a standard network represent connectivity, providing information about how one node can be reached from another. Dependency links represent a need for support from one node to another. This relationship is often, though not necessarily, mutual and thus the links can be directed or undirected. Crucially, a node loses its ability to function as soon as the node it is dependent on ceases to function while it may not be so severely effected by losing a node it is connected to.

In graph theory, a strong orientation of an undirected graph is an assignment of a direction to each edge (an orientation) that makes it into a strongly connected graph. Strong orientations have been applied to the design of one- way road networks. According to Robbins' theorem, the graphs with strong orientations are exactly the bridgeless graphs. Eulerian orientations and well-balanced orientations provide important special cases of strong orientations; in turn, strong orientations may be generalized to totally cyclic orientations of disconnected graphs.

A tree is a connected undirected graph with no cycles. It is a spanning tree of a graph G if it spans G (that is, it includes every vertex of G) and is a subgraph of G (every edge in the tree belongs to G). A spanning tree of a connected graph G can also be defined as a maximal set of edges of G that contains no cycle, or as a minimal set of edges that connect all vertices.

In graph theory, Schnyder's theorem is a characterization of planar graphs in terms of the order dimension of their incidence posets. It is named after Walter Schnyder, who published its proof in 1989. The incidence poset of an undirected graph with vertex set and edge set is the partially ordered set of height 2 that has as its elements. In this partial order, there is an order relation when is a vertex, is an edge, and is one of the two endpoints of .

Pósa's theorem, in graph theory, is a sufficient condition for the existence of a Hamiltonian cycle based on the degrees of the vertices in an undirected graph. It implies two other degree-based sufficient conditions, Dirac's theorem on Hamiltonian cycles and Ore's theorem. Unlike those conditions, it can be applied to graphs with a small number of low-degree vertices. It is named after Lajos Pósa, a protégé of Paul Erdős born in 1947, who discovered this theorem in 1962.

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.

A standard example for a kernelization algorithm is the kernelization of the vertex cover problem by S. Buss.This unpublished observation is acknowledged in a paper of In this problem, the input is an undirected graph G together with a number k. The output is a set of at most k vertices that includes an endpoint of every edge in the graph, if such a set exists, or a failure exception if no such set exists. This problem is NP-hard.

In the mathematical area of graph theory, an undirected graph G is dually chordal if the hypergraph of its maximal cliques is a hypertree. The name comes from the fact that a graph is chordal if and only if the hypergraph of its maximal cliques is the dual of a hypertree. Originally, these graphs were defined by maximum neighborhood orderings and have a variety of different characterizations.; ; ; ; Unlike for chordal graphs, the property of being dually chordal is not hereditary, i.e.

A planar graph and its minimum spanning tree. Each edge is labeled with its weight, which here is roughly proportional to its length. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible.

Any undirected graph may be made into a DAG by choosing a total order for its vertices and directing every edge from the earlier endpoint in the order to the later endpoint. The resulting orientation of the edges is called an acyclic orientation. Different total orders may lead to the same acyclic orientation, so an -vertex graph can have fewer than acyclic orientations. The number of acyclic orientations is equal to , where is the chromatic polynomial of the given graph.. condensation of the blue directed graph.

In graph theory, a random geometric graph (RGG) is the mathematically simplest spatial network, namely an undirected graph constructed by randomly placing N nodes in some metric space (according to a specified probability distribution) and connecting two nodes by a link if and only if their distance is in a given range, e.g. smaller than a certain neighborhood radius, r. Random geometric graphs resemble real human social networks in a number of ways. For instance, they spontaneously demonstrate community structure - clusters of nodes with high modularity.

The minimum cut problem in undirected, weighted graphs can be solved in polynomial time by the Stoer-Wagner algorithm. In the special case when the graph is unweighted, Karger's algorithm provides an efficient randomized method for finding the cut. In this case, the minimum cut equals the edge connectivity of the graph. A generalization of the minimum cut problem without terminals is the minimum -cut, in which the goal is to partition the graph into at least connected components by removing as few edges as possible.

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.

A clique, C, in an undirected graph is a subset of the vertices, , such that every two distinct vertices are adjacent. This is equivalent to the condition that the induced subgraph of G induced by C is a complete graph. In some cases, the term clique may also refer to the subgraph directly. A maximal clique is a clique that cannot be extended by including one more adjacent vertex, that is, a clique which does not exist exclusively within the vertex set of a larger clique.

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.

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.

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.

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).

In the case of unweighted graphs the calculations can be done with Brandes' algorithm which takes O(V E) time. Normally, these algorithms assume that graphs are undirected and connected with the allowance of loops and multiple edges. When specifically dealing with network graphs, often graphs are without loops or multiple edges to maintain simple relationships (where edges represent connections between two people or vertices). In this case, using Brandes' algorithm will divide final centrality scores by 2 to account for each shortest path being counted twice.

In the vertical dimension, the teacher would provide and order questions aimed at the development of understanding ideas (not for covering predetermined ground). In a horizontal dimension, discussion would be open to all possible answers from students in response to the questions. If a seminar is too open in both dimensions, or focused primarily within the horizontal dimension, it may become loose and undirected. When it is directed and controlled in both dimensions or focused primarily on the vertical dimension, it becomes didactic and dogmatic.

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.

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.

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.

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.

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 .

Social collaboration is related to social networking, with the distinction that while social networking is individual- centric, social collaboration is entirely group-centric. Generally speaking, social networking means socializing for personal, professional or entertainment purposes, for example, LinkedIn and Facebook. Social collaboration, on the other hand, means working socially to achieve a common goal, for example, GitHub and Quora. Social networking services generally focus on individuals sharing messages in a more-or-less undirected way and receiving messages from many sources into a single personalized activity feed.

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.

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.

In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. The triangle-free graphs with the most edges for their vertices are balanced complete bipartite graphs. Many triangle-free graphs are not bipartite, for example any cycle graph Cn for odd n > 3\.

After his father's death in 1901 Spengler attended several universities (Munich, Berlin, and Halle) as a private scholar, taking courses in a wide range of subjects. His private studies were undirected. In 1903, he failed his doctoral thesis on Heraclitus (titled Der metaphysische Grundgedanke der heraklitischen Philosophie (The Fundamental Metaphysical Thought of the Heraclitean Philosophy) and conducted under the direction of Alois Riehl) because of insufficient references. He eventually took the doctoral oral exam again and received his PhD from Halle on 6 April 1904.

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.

In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root.. See p. 454. Both directed and undirected versions of rooted graphs have been studied, and there are also variant definitions that allow multiple roots. Rooted graphs may also be known (depending on their application) as pointed graphs or flow graphs. In some of the applications of these graphs, there is an additional requirement that the whole graph be reachable from the root vertex.

In mathematics, a rotation map is a function that represents an undirected edge-labeled graph, where each vertex enumerates its outgoing neighbors. Rotation maps were first introduced by Reingold, Vadhan and Wigderson (“Entropy waves, the zig-zag graph product, and new constant-degree expanders”, 2002) in order to conveniently define the zig-zag product and prove its properties. Given a vertex v and an edge label i, the rotation map returns the i'th neighbor of v and the edge label that would lead back to v.

In his 1736 paper on the Seven Bridges of Königsberg, widely considered to be the birth of graph theory, Leonhard Euler proved that, for a finite undirected graph to have a closed walk that visits each edge exactly once, it is necessary and sufficient that it be connected except for isolated vertices (that is, all edges are contained in one component) and have even degree at each vertex. The corresponding characterization for the existence of a closed walk visiting each edge exactly once in a directed graph is that the graph be strongly connected and have equal numbers of incoming and outgoing edges at each vertex. In either case, the resulting walk is known as an Euler cycle or Euler tour. If a finite undirected graph has even degree at each of its vertices, regardless of whether it is connected, then it is possible to find a set of simple cycles that together cover each edge exactly once: this is Veblen's theorem.. When a connected graph does not meet the conditions of Euler's theorem, a closed walk of minimum length covering each edge at least once can nevertheless be found in polynomial time by solving the route inspection problem.

Dembski describes specified complexity as a property in living things which can be observed by intelligent-design proponents. However, whereas Orgel used the term for biological features which are considered in science to have arisen through a process of evolution, Dembski says that it describes features which cannot form through "undirected" evolution--and concludes that it allows one to infer intelligent design. While Orgel employed the concept in a qualitative way, Dembski's use is intended to be quantitative. Dembski's use of the concept dates to his 1998 monograph The Design Inference.

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.

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.

In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. uses a reversed terminology, in which he called bond matroids "graphic" and cycle matroids "co-graphic", but this has not been followed by later authors. A matroid that is both graphic and co-graphic is called a planar matroid; these are exactly the graphic matroids formed from planar graphs.

Other biological effects and feedbacks exist, but the extent to which these mechanisms have stabilized and modified the Earth's overall climate is largely not known. The Gaia hypothesis is sometimes viewed from significantly different philosophical perspectives. Some environmentalists view it as an almost conscious process, in which the Earth's ecosystem is literally viewed as a single unified organism. Some evolutionary biologists, on the other hand, view it as an undirected emergent property of the ecosystem: as each individual species pursues its own self-interest, their combined actions tend to have counterbalancing effects on environmental change.

The white vertex sets are maximal nonblockers In graph theory, a nonblocker is a subset of vertices in an undirected graph, all of which are adjacent to vertices outside of the subset. Equivalently, a nonblocker is the complement of a dominating set. The computational problem of finding the largest nonblocker in a graph was formulated by , who observed that it belongs to MaxSNP. Although computing a dominating set is not fixed-parameter tractable under standard assumptions, the complementary problem of finding a nonblocker of a given size is fixed-parameter tractable.

Although an avid reader since childhood, Rozgonyi did not begin writing until late 2001. Undirected and uncertain in his path, he dropped out of university and drifted through a variety of fields and interests in which he achieved limited success. These included playing the piano and alto saxophone in a variety of small jazz and blues ensembles, and billiards, earning 2nd Place in the international VNEA World Artistic Pool Challenge in Las Vegas, Nevada (1997). He turned to travel in the same year, and began collecting the experiences that would later inform his work.

Plantinga's "proper function" account argues that as a necessary condition of having warrant, one's "belief-forming and belief-maintaining apparatus of powers" are functioning properly—"working the way it ought to work".WPF, p. 4 Plantinga explains his argument for proper function with reference to a "design plan", as well as an environment in which one's cognitive equipment is optimal for use. Plantinga asserts that the design plan does not require a designer: "it is perhaps possible that evolution (undirected by God or anyone else) has somehow furnished us with our design plans",WPF, p.

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 .

WPF, p. 4 Plantinga explains his argument for proper function with reference to a "design plan", as well as an environment in which one's cognitive equipment is optimal for use. Plantinga asserts that the design plan does not require a designer: "it is perhaps possible that evolution (undirected by God or anyone else) has somehow furnished us with our design plans",WPF, p. 21 but the paradigm case of a design plan is like a technological product designed by a human being (like a radio or a wheel).

Testing whether an arbitrary graph has a Hamiltonian decomposition is NP-complete, both in the directed and undirected cases. The line graphs of cubic graphs are 4-regular, and have a Hamiltonian decomposition if and only if the underlying cubic graph has a Hamiltonian cycle. As a consequence, Hamiltonian decomposition remains NP-complete for classes of graphs that include line graphs of hard instances of the Hamiltonian cycle problem. For instance, Hamiltonian decomposition is NP- complete for the 4-regular planar graphs, because they include the line graphs of cubic planar graphs.

It is undecidable whether a given first-order sentence can be realized by a finite undirected graph. writes that this undecidability result is well known, and attributes it to on the undecidability of first order satisfiability for more general classes of finite structures. There exist first-order sentences that are modeled by infinite graphs but not by any finite graph. For instance, the property of having exactly one vertex of degree one, with all other vertices having degree exactly two, can be expressed by a first order sentence.

A min-cut of a weighted graph having min-cut weight 4 In graph theory, the Stoer–Wagner algorithm is a recursive algorithm to solve the minimum cut problem in undirected weighted graphs with non-negative weights. It was proposed by Mechthild Stoer and Frank Wagner in 1995. The essential idea of this algorithm is to shrink the graph by merging the most intensive vertices, until the graph only contains two combined vertex sets. At each phase, the algorithm finds the minimum s-t cut for two vertices s and t chosen at its will.

The constraint composite graph is a node-weighted undirected graph associated with a given combinatorial optimization problem posed as a weighted constraint satisfaction problem. Developed and introduced by Satish Kumar Thittamaranahalli (T. K. Satish Kumar), the idea of the constraint composite graph is a big step towards unifying different approaches for exploiting "structure" in weighted constraint satisfaction problems.Kumar, T. K. S. (2008), "A Framework for Hybrid Tractability Results in Boolean Weighted Constraint Satisfaction Problems", Proceedings of the Fourteenth International Conference on Principles and Practice of Constraint Programming (CP).

A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph. That is, it is an orientation of a complete graph, or equivalently a directed graph in which every pair of distinct vertices is connected by a directed edge with any one of the two possible orientations. Many of the important properties of tournaments were first investigated by in order to model dominance relations in flocks of chickens. Current applications of tournaments include the study of voting theory and social choice theory among other things.

Chinese Whispers is a hard partitioning, randomized, flat clustering (no hierarchical relations between clusters) method. The random property means that running the process on the same network several times can lead to different results, while because of hard partitioning one node can only belong to one cluster at a given moment. The original algorithm is applicable to undirected, weighted and unweighted graphs. Chinese Whispers is time linear which means that it is extremely fast even if the number of nodes and links are very high in the network.

The all-ones matrix arises in the mathematical field of combinatorics, particularly involving the application of algebraic methods to graph theory. For example, if A is the adjacency matrix of a n-vertex undirected graph G, and J is the all-ones matrix of the same dimension, then G is a regular graph if and only if AJ = JA.. As a second example, the matrix appears in some linear-algebraic proofs of Cayley's formula, which gives the number of spanning trees of a complete graph, using the matrix tree theorem.

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.

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.

Another approach to collective classification is to represent the problem with a graphical model and use learning and inference techniques for the graphical modeling approach to arrive at the correct classifications. Graphical models are tools for joint, probabilistic inference, making them ideal for collective classification. They are characterized by a graphical representation of a probability distribution P, in which random variables are nodes in a graph G. Graphical models can be broadly categorized by whether the underlying graph is directed (e.g., Bayesian networks or collections of local classifiers) or undirected (e.g.

The undirected capacitated arc routing problem consists of demands placed on the edges, and each edge must meet the demand. An example is garbage collection, where each route might require both a garbage collection and a recyclable collection. Problems in real life applications might arise if there are timing issues, such as the case in which certain routes cannot be serviced due to timing or scheduling conflicts, or constraints, such as a limited period of time. The heuristics described in this article ignore any such problems that arise due to application constraints.

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.

Origin The first chapter of part II, "Darwinian Thinking in Biology", asserts that life originated without any skyhooks, and the orderly world we know is the result of a blind and undirected shuffle through chaos. The eighth chapter's message is conveyed by its title, "Biology is Engineering"; biology is the study of design, function, construction and operation. However, there are some important differences between biology and engineering. Related to the engineering concept of optimization, the next chapter deals with adaptationism, which Dennett endorses, calling Gould and Lewontin's "refutation" of it an illusion.

4 Plantinga explains his argument for proper function with reference to a "design plan", as well as an environment in which one's cognitive equipment is optimal for use. Plantinga asserts that the design plan does not require a designer: "it is perhaps possible that evolution (undirected by God or anyone else) has somehow furnished us with our design plans",WPF, p. 21 but the paradigm case of a design plan is like a technological product designed by a human being (like a radio or a wheel). Ultimately, Plantinga argues that epistemological naturalism- i.e.

For keyphrase extraction, it builds a graph using some set of text units as vertices. Edges are based on some measure of semantic or lexical similarity between the text unit vertices. Unlike PageRank, the edges are typically undirected and can be weighted to reflect a degree of similarity. Once the graph is constructed, it is used to form a stochastic matrix, combined with a damping factor (as in the "random surfer model"), and the ranking over vertices is obtained by finding the eigenvector corresponding to eigenvalue 1 (i.e.

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.

A demo for Prim's algorithm based on Euclidean distance. In computer science, Prim's (also known as Jarník's) algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. The algorithm operates by building this tree one vertex at a time, from an arbitrary starting vertex, at each step adding the cheapest possible connection from the tree to another vertex.

Branch decomposition of a grid graph, showing an e-separation. The separation, the decomposition, and the graph all have width three. In graph theory, a branch-decomposition of an undirected graph G is a hierarchical clustering of the edges of G, represented by an unrooted binary tree T with the edges of G as its leaves. Removing any edge from T partitions the edges of G into two subgraphs, and the width of the decomposition is the maximum number of shared vertices of any pair of subgraphs formed in this way.

The distinction between acquisition and learning can be used in this discussion, because the general conditions in the case of second language offer opportunities for acquisition, because it is informal, free, undirected or naturalistic. On the other hand, educational treatment in the case of foreign language may offer opportunities mainly for learning. Nevertheless, acquisition can take place in the case of foreign language learning and learning can take place in the case of second language learning. For example, immigrants to the US can attend language teaching classes in the target language environment.

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.

Due to cone photoreceptor damage located in the macula, there is a significant reduction of visual input to the visual association cortex, stirring endogenous activation in the color areas and thus leading to colored hallucinations. Patients with CBS alongside macular degeneration exhibit hyperactivity in the color areas of the visual association cortex (as shown in fMRI’s). Those with significant ocular disease yet maintain visual acuity may still be susceptible to CBS. The Deep Boltzmann Machine (DBM) is a way of utilizing an undirected probabilistic process in a neural framework.

Of all 'events' studied by Project Hindsight, 91% were technological, and only 9% were classed as science. Within the latter category 8.7% were applied science, whereas only 0.3%, or two 'events', were due to basic or undirected science.D. S. Greenberg, 'Hindsight: DOD Study Examines Return on Investment in Research' Science 154 (November 18, 1966): 872-73Philip H. Abelson, 'Project Hindsight,' Science 154 (December 2, 1966): 1123. This particular finding undermined the traditional view that technological progress is the outcome of basic research since the direct influence of science on technology is very small.

Mikhail Vartanov (, , ; b. February 21, 1937, RSFSR, Soviet Union, now Russian Federation, d. December 31, 2009, Hollywood, California) was a film director, cinematographer, documentarian, essayist, photographer and artist who developed a style of documentary filmmaking termed the "direction of undirected action." He is considered an important cinematographer and documentarian of his generation, noted for artistic collaborations with Sergei Parajanov and such influential documentary films as Parajanov: The Last Spring, The Seasons (directed by Artavazd Peleshyan), The Color of Armenian Land, and a series of essays including The Unmailed Letters.

While the parameter k in the examples above is chosen as the size of the desired solution, this is not necessary. It is also possible to choose a structural complexity measure of the input as the parameter value, leading to so-called structural parameterizations. This approach is fruitful for instances whose solution size is large, but for which some other complexity measure is bounded. For example, the feedback vertex number of an undirected graph G is defined as the minimum cardinality of a set of vertices whose removal makes G acyclic.

An oriented graph is a finite directed graph obtained from a simple undirected graph by assigning an orientation to each edge. Equivalently, it is a directed graph that has no self-loops, no parallel edges, and no two-edge cycles. The first neighborhood of a vertex v (also called its open neighborhood) consists of all vertices at distance one from v, and the second neighborhood of v consists of all vertices at distance two from v. These two neighborhoods form disjoint sets, neither of which contains v itself.

By Menger's theorem, for any two vertices and in a connected graph , the numbers and can be determined efficiently using the max-flow min-cut algorithm. The connectivity and edge-connectivity of can then be computed as the minimum values of and , respectively. In computational complexity theory, SL is the class of problems log-space reducible to the problem of determining whether two vertices in a graph are connected, which was proved to be equal to L by Omer Reingold in 2004. Hence, undirected graph connectivity may be solved in space.

Many of these generalized notions of cliques can also be found by constructing an undirected graph whose edges represent related pairs of actors from the social network, and then applying an algorithm for the clique problem to this graph. Since the work of Harary and Ross, many others have devised algorithms for various versions of the clique problem. In the 1970s, researchers began studying these algorithms from the point of view of worst-case analysis. See, for instance, , an early work on the worst-case complexity of the maximum clique problem.

During the 1960s, the Defense Advanced Research Projects Agency (then known as "ARPA", now known as "DARPA") provided millions of dollars for AI research with almost no strings attached. DARPA's director in those years, J. C. R. Licklider believed in "funding people, not projects" and allowed AI's leaders (such as Marvin Minsky, John McCarthy, Herbert A. Simon or Allen Newell) to spend it almost any way they liked. This attitude changed after the passage of Mansfield Amendment in 1969, which required DARPA to fund "mission-oriented direct research, rather than basic undirected research". (only the sections before 1980 apply to the current discussion).

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.

Illuminating the skeleton of a convex polyhedron from a light source close to one of its faces causes its shadows to form a planar Schlegel diagram. An undirected graph is a system of vertices and edges, each edge connecting two of the vertices. From any polyhedron one can form a graph, by letting the vertices of the graph correspond to the vertices of the polyhedron and by connecting any two graph vertices by an edge whenever the corresponding two polyhedron vertices are the endpoints of an edge of the polyhedron. This graph is known as the skeleton of the polyhedron.

In graph theory, a branch of mathematics and computer science, the Chinese postman problem, postman tour or route inspection problem is to find a shortest closed path or circuit that visits every edge of a (connected) undirected graph. When the graph has an Eulerian circuit (a closed walk that covers every edge once), that circuit is an optimal solution. Otherwise, the optimization problem is to find the smallest number of graph edges to duplicate (or the subset of edges with the minimum possible total weight) so that the resulting multigraph does have an Eulerian circuit. It can be solved in polynomial time.

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.

GQL graphs can be mixed: they can contain directed edges, where one of the endpoint nodes of an edge is the tail (or source) and the other node is the head (or target or destination), but they can also contain undirected (bidirectional or reflexive) edges. Nodes and edges, collectively known as elements, have attributes. Those attributes may be data values, or labels (tags). Values of properties cannot be elements of graphs, nor can they be whole graphs: these restrictions intentionally force a clean separation between the topology of a graph, and the attributes carrying data values in the context of a graph topology.

In the mathematics of structural rigidity, a rigidity matroid is a matroid that describes the number of degrees of freedom of an undirected graph with rigid edges of fixed lengths, embedded into Euclidean space. In a rigidity matroid for a graph with n vertices in d-dimensional space, a set of edges that defines a subgraph with k degrees of freedom has matroid rank dn − k. A set of edges is independent if and only if, for every edge in the set, removing the edge would increase the number of degrees of freedom of the remaining subgraph....

In graph theory, the cycle rank of a directed graph is a digraph connectivity measure proposed first by Eggan and Büchi . Intuitively, this concept measures how close a digraph is to a directed acyclic graph (DAG), in the sense that a DAG has cycle rank zero, while a complete digraph of order n with a self-loop at each vertex has cycle rank n. The cycle rank of a directed graph is closely related to the tree-depth of an undirected graph and to the star height of a regular language. It has also found use in sparse matrix computations (see ) and logic .

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.

The agencies which funded AI research (such as the British government, DARPA and NRC) became frustrated with the lack of progress and eventually cut off almost all funding for undirected research into AI. The pattern began as early as 1966 when the ALPAC report appeared criticizing machine translation efforts. After spending 20 million dollars, the NRC ended all support. , , and under "Success in Speech Recognition". In 1973, the Lighthill report on the state of AI research in England criticized the utter failure of AI to achieve its "grandiose objectives" and led to the dismantling of AI research in that country.

A 2-degenerate graph: each vertex has at most two neighbors to its left, so the rightmost vertex of any subgraph has degree at most two. Its 2-core, the subgraph remaining after repeatedly deleting vertices of degree less than two, is shaded. In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate.

In London in June 2005, Ellams founded The Midnight Run, an undirected and politically-unaffiliated urban social excursion project where strangers gather to explore a city’s streets from 6pm to midnight or 6am. The project originated when Ellams and a close friend were waiting one night for a bus after seeing a play at Battersea Arts Centre. The bus did not show up, so they walked the bus route aimlessly for hours discovering London. Further influenced by the 1950s French artistic movement The Situationists and the nomadic traditions of Ellams' Nigerian Hausa tribe, Ellams began inviting members of his poetry newsletter mailing list.

In the context of this article, all graphs will be simple and undirected, unless stated otherwise. This means that the edges of the graph form a set (and not a multiset) and each edge is a pair of distinct vertices. Graphs are assumed to have an implicit representation in which each vertex has a unique identifier or label and in which it is possible to test the adjacency of any two vertices, but for which adjacency testing is the only allowed primitive operation. Informally, a graph property is a property of a graph that is independent of labeling.

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..

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.

Every 4-regular undirected graph has an even number of Hamiltonian decompositions. More strongly, for every two edges e and f of a 4-regular graph, the number of Hamiltonian decompositions in which e and f belong to the same cycle is even. If a 2k-regular graph has a Hamiltonian decomposition, it has at least a triple factorial number of decompositions, :(3k-2)\cdot(3k-5)\cdots 7\cdot 4 \cdot 1. For instance, 4-regular graphs that have a Hamiltonian decomposition have at least four of them; 6-regular graphs that have a Hamiltonian decomposition have at least 28, etc.

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.

The main idea of the proof is to observe that the Cayley graph of G, with the addition of colors and orientations on its edges to distinguish the generators of G from each other, has the desired automorphism group. Therefore, if each of these edges is replaced by an appropriate subgraph, such that each replacement subgraph is itself asymmetric and two replacements are isomorphic if and only if they replace edges of the same color, then the undirected graph created by performing these replacements will also have G as its symmetry group., discussion following Theorem 4.1.

The Levi graphs of projective configurations lead to many important symmetric graphs and cages. The visibility graph of a closed polygon connects each pair of vertices by an edge whenever the line segment connecting the vertices lies entirely in the polygon. It is not known how to test efficiently whether an undirected graph can be represented as a visibility graph. A partial cube is a graph for which the vertices can be associated with the vertices of a hypercube, in such a way that distance in the graph equals Hamming distance between the corresponding hypercube vertices.

However, adults tend to (often mistakenly) assume that virtually all children's social activities can be understood as "play" and, furthermore, that children's play activities do not involve much skill or effort. It is through play that children at a very early age engage and interact in the world around them. Play allows children to create and explore a world they can master, conquering their fears while practicing adult roles, sometimes in conjunction with other children or adult caregivers. Undirected play allows children to learn how to work in groups, to share, to negotiate, to resolve conflicts, and to learn self-advocacy skills.

An undirected graph colored based on the betweenness centrality of each vertex from least (red) to greatest (blue). In graph theory, betweenness centrality is a measure of centrality in a graph based on shortest paths. For every pair of vertices in a connected graph, there exists at least one shortest path between the vertices such that either the number of edges that the path passes through (for unweighted graphs) or the sum of the weights of the edges (for weighted graphs) is minimized. The betweenness centrality for each vertex is the number of these shortest paths that pass through the vertex.

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).

It was invented by David Karger and first published in 1993. The idea of the algorithm is based on the concept of contraction of an edge (u, v) in an undirected graph G = (V, E). Informally speaking, the contraction of an edge merges the nodes u and v into one, reducing the total number of nodes of the graph by one. All other edges connecting either u or v are "reattached" to the merged node, effectively producing a multigraph. Karger's basic algorithm iteratively contracts randomly chosen edges until only two nodes remain; those nodes represent a cut in the original graph.

The relative neighborhood graph of 100 random points in a unit square. In computational geometry, the relative neighborhood graph (RNG) is an undirected graph defined on a set of points in the Euclidean plane by connecting two points p and q by an edge whenever there does not exist a third point r that is closer to both p and q than they are to each other. This graph was proposed by Godfried Toussaint in 1980 as a way of defining a structure from a set of points that would match human perceptions of the shape of the set...

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.

In graph theory, a Pfaffian orientation of an undirected graph G is an orientation (an assignment of a direction to each edge of the graph) in which every even central cycle is oddly oriented. In this definition, a cycle C is even if it contains an even number of edges. C is central if the subgraph of G formed by removing all the vertices of C has a perfect matching; central cycles are also sometimes called alternating circuits. And C is oddly oriented if each of the two orientations of C is consistent with an odd number of edges in the orientation.

The Towers of Hanoi puzzle involves moving disks of different sizes between three pegs, maintaining the property that no disk is ever placed on top of a smaller disk. The states of an n-disk puzzle, and the allowable moves from one state to another, form an undirected graph, the Hanoi graph, that can be represented geometrically as the intersection graph of the set of triangles remaining after the nth step in the construction of the Sierpinski triangle. Thus, in the limit as n goes to infinity, this sequence of graphs can be interpreted as a discrete analogue of the Sierpinski triangle..

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,.

Gaussian prime numbers whose minimax path length is 2 or more. A variant of the minimax path problem has also been considered for sets of points in the Euclidean plane. As in the undirected graph problem, this Euclidean minimax path problem can be solved efficiently by finding a Euclidean minimum spanning tree: every path in the tree is a minimax path. However, the problem becomes more complicated when a path is desired that not only minimizes the hop length but also, among paths with the same hop length, minimizes or approximately minimizes the total length of the path.

If the input digraphs are restricted to be tournaments, the resulting problem is known as the minimum feedback arc set problem on tournaments (FAST). This restricted problem does admit a polynomial-time approximation scheme, and this still holds for a restricted weighted version of the problem.. See also author's extended version. A subexponential fixed parameter algorithm for the weighted FAST was given by .. On the other hand, if the edges are undirected, the problem of deleting edges to make the graph cycle-free is equivalent to finding a minimum spanning tree, which can be done easily in polynomial time.

If a classification task is to separate pictures of cats and dogs then a model of this kind will only be able to decide whether a picture is of a cat or a dog. This is decided according to the most similar example from the training data (see supervised learning). A generative model on the other hand will be able to produce a new picture of a either class. Typical discriminative models include logistic regression (LR), support vector machines (SVM), conditional random fields (CRFs) (specified over an undirected graph), decision trees, neural networks, and many others.

The Kleinberg model of a network is effective at demonstrating the effectiveness of greedy small world routing. The model uses an n x n grid of nodes to represent a network, where each node is connected with an undirected edge to its neighbors. To give it the “small world” effect, a number of long range edges are added to the network that tend to favor nodes closer in distance rather than farther. When adding edges, the probability of connecting some random vertex v to another random vertex w is proportional to 1/d(v,w)^q, where q is the clustering exponent.

A Hamiltonian cycle around a network of six vertices In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron.

In this graph, triangle 1-2-5 is convex, but path 2-3-4 is not, because it does not include one of the two shortest paths from 2 to 4. In metric graph theory, a convex subgraph of an undirected graph G is a subgraph that includes every shortest path in G between two of its vertices. Thus, it is analogous to the definition of a convex set in geometry, a set that contains the line segment between every pair of its points. Convex subgraphs play an important role in the theory of partial cubes and median graphs.

In graph theory, a haven is a certain type of function on sets of vertices in an undirected graph. If a haven exists, it can be used by an evader to win a pursuit-evasion game on the graph, by consulting the function at each step of the game to determine a safe set of vertices to move into. Havens were first introduced by as a tool for characterizing the treewidth of graphs. Their other applications include proving the existence of small separators on minor- closed families of graphs, and characterizing the ends and clique minors of infinite graphs...

In computational geometry, a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects. Computing the convex hull means constructing an unambiguous, efficient representation of the required convex shape. Output representations that have been considered for convex hulls of point sets include a list of linear inequalities describing the facets of the hull, an undirected graph of facets and their adjacencies, or the full face lattice of the hull. In two dimensions, it may suffice more simply to list the points that are vertices, in their cyclic order around the hull.

Moreover, it can be inferred from the results in that using the symmetry-breaking conditions results in high efficiency particularly for directed networks in comparison to undirected networks. The symmetry-breaking conditions used in the GK algorithm are similar to the restriction which ESU algorithm applies to the labels in EXT and SUB sets. In conclusion, the GK algorithm computes the exact number of appearance of a given query graph in a large complex network and exploiting symmetry-breaking conditions improves the algorithm performance. Also, GK algorithm is one of the known algorithms having no limitation for motif size in implementation and potentially it can find motifs of any size.

The total symmetry factor is 2, and the contribution of this diagram is divided by 2. The symmetry factor theorem gives the symmetry factor for a general diagram: the contribution of each Feynman diagram must be divided by the order of its group of automorphisms, the number of symmetries that it has. An automorphism of a Feynman graph is a permutation of the lines and a permutation of the vertices with the following properties: # If a line goes from vertex to vertex , then goes from to . If the line is undirected, as it is for a real scalar field, then can go from to too.

A nearest neighbor graph of 100 points in the Euclidean plane. The nearest neighbor graph (NNG) for a set of n objects P in a metric space (e.g., for a set of points in the plane with Euclidean distance) is a directed graph with P being its vertex set and with a directed edge from p to q whenever q is a nearest neighbor of p (i.e., the distance from p to q is no larger than from p to any other object from P). In many discussions the directions of the edges are ignored and the NNG is defined as an ordinary (undirected) graph.

Dragon View is a side-scrolling role-playing beat 'em up released for the Super Nintendo Entertainment System in November 1994. Released in Japan as and otherwise known as Drakkhen II, it is meant to be a sequel to Drakkhen although it bears little resemblance to its predecessor. It uses the same pseudo-3D overworld system for which the series is most famous. Other features of Dragon View are its side-view action role-playing game (RPG) hybrid gameplay (used when exploring more detailed areas such as towns and dungeons), it's well translated first-person storyline, and its emphasis on player-driven undirected exploration.

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.

In a complete undirected graph G = (V, E), if we sort the edges in nondecreasing order of the distances: d(e1) ≤ d(e2) ≤ … ≤ d(em) and let Gi = (V, Ei), where Ei = {e1, e2, …, ei}. The k-center problem is equivalent to finding the smallest index i such that Gi has a dominating set of size at most k. Although Dominating Set is NP-complete, the k-center problem remains NP- hard. This is clear, since the optimality of a given feasible solution for the k-center problem can be determined through the Dominating Set reduction only if we know in first place the size of the optimal solution (i.e.

In graph theory, a universal vertex is a vertex of an undirected graph that is adjacent to all other vertices of the graph. It may also be called a dominating vertex, as it forms a one-element dominating set in the graph. (It is not to be confused with a universally quantified vertex in the logic of graphs.) A graph that contains a universal vertex may be called a cone. In this context, the universal vertex may also be called the apex of the cone.. However, this terminology conflicts with the terminology of apex graphs, in which an apex is a vertex whose removal leaves a planar subgraph.

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.

SSPACE(S(n)) is the class of the languages accepted by a symmetric Turing machine running in space O(S(n)) and SL=SSPACE(log(n)). SL can equivalently be defined as the class of problems logspace reducible to USTCON. Lewis and Papadimitriou by their definition showed this by constructing a nondeterministic machine for USTCON with properties that they showed are sufficient to make a construction of an equivalent symmetric Turing machine possible. Then, they observed that any language in SL is logspace reducible to USTCON as from the properties of the symmetric computation we can view the special configuration as the undirected edges of the graph.

Given an undirected graph with non-negative edge weights and a subset of vertices, usually referred to as terminals, the Steiner tree problem in graphs requires a tree of minimum weight that contains all terminals (but may include additional vertices). Further well-known variants are the Euclidean Steiner tree problem and the rectilinear minimum Steiner tree problem. The Steiner tree problem in graphs can be seen as a generalization of two other famous combinatorial optimization problems: the (non-negative) shortest path problem and the minimum spanning tree problem. If a Steiner tree problem in graphs contains exactly two terminals, it reduces to finding the shortest path.

Seidel's algorithm is an algorithm designed by Raimund Seidel in 1992 for the all-pairs-shortest-path problem for undirected, unweighted, connected graphs. It solves the problem in O(V^\omega \log V) expected time for a graph with V vertices, where \omega < 2.373 is the exponent in the complexity O(n^\omega) of n \times n matrix multiplication. If only the distances between each pair of vertices are sought, the same time bound can be achieved in the worst case. Even though the algorithm is designed for connected graphs, it can be applied individually to each connected component of a graph with the same running time overall.

The discrepant behaviors are most often exhibited on reunion, but are found in other episodes of the procedure as well. Main and Cassidy developed a set of thematic headings for the various forms of disorganized/disoriented behavior. Infant behaviors coded as disorganized/disoriented include sequential display of contradictory behavior patterns (Index I); simultaneous display of contradictory behavior patterns (II); undirected, misdirected, incomplete, and interrupted movements and expressions (III); stereotypies, asymmetrical movements, mistimed movements, and anomalous postures (IV); freezing, stilling, and slowed movements and expressions (V); direct indices of apprehension regarding the parent (VI); direct indices of disorganization or disorientation (VII).Main, M., & Solomon, J. (1990).

None of these letters has been located to date. Together with Craig's published work, in particular his 1918 essay on appetites and aversions, they were regarded by Lorenz as foundational for the development of ethology. A key insight was that much behavior is expressed not in response to, but in search of sensory input – or the lack of such input. With increasing appetite, animals engage in an undirected search for food, and only once located will the food stimuli be tracked down and consumption ensue. The role of learning in these processes is to provide ever more ‘educated guesses’ during the initial search phase.

The classic sequential algorithm for computing biconnected components in a connected undirected graph is due to John Hopcroft and Robert Tarjan (1973). It runs in linear time, and is based on depth-first search. This algorithm is also outlined as Problem 22-2 of Introduction to Algorithms (both 2nd and 3rd editions). The idea is to run a depth-first search while maintaining the following information: # the depth of each vertex in the depth-first-search tree (once it gets visited), and # for each vertex v, the lowest depth of neighbors of all descendants of v (including v itself) in the depth-first-search tree, called the lowpoint.

In graph theory, the tree-depth of a connected undirected graph G is a numerical invariant of G, the minimum height of a Trémaux tree for a supergraph of G. This invariant and its close relatives have gone under many different names in the literature, including vertex ranking number, ordered chromatic number, and minimum elimination tree height; it is also closely related to the cycle rank of directed graphs and the star height of regular languages.; ; , p. 116. Intuitively, where the treewidth graph width parameter measures how far a graph is from being a tree, this parameter measures how far a graph is from being a star.

Computing tree-depth is computationally hard: the corresponding decision problem is NP-complete.. The problem remains NP-complete for bipartite graphs , as well as for chordal graphs.. On the positive side, tree-depth can be computed in polynomial time on interval graphs,. as well as on permutation, trapezoid, circular-arc, circular permutation graphs, and cocomparability graphs of bounded dimension.. For undirected trees, tree-depth can be computed in linear time.; . give an approximation algorithm for tree-depth with an approximation ratio of O((\log n)^2), based on the fact that tree-depth is always within a logarithmic factor of the treewidth of a graph.

ARAG members were hand-picked and came from a wide background of skills, experience and nationalities. ARAG subsumed the Conflict Studies Research Centre, previously known as the Soviet Studies Research Centre, which for nearly 40 years had studied the Soviet Union, working in open source organic languages to supplement MoD and Intelligence Service assets. ARAG worked across government departments undertaking both directed and undirected research and broadening its reach to encompass new security challenges such as China, cyber security and strategic communication. It produced a large number of research papers - made freely available to the public from the Defence Academy website - as well as more targeted limited distribution work.

The concessions came hand-in-hand with renewed, and brutal, action against the unrest. There was also a backlash from the conservative elements of society, with right-wing attacks on strikers, left-wingers, and Jews. While the Russian liberals were satisfied by the October Manifesto and prepared for upcoming Duma elections, radical socialists and revolutionaries denounced the elections and called for an armed uprising to destroy the Empire. A locomotive overturned by striking workers at the main railway depot in Tiflis in 1905 Some of the November uprising of 1905 in Sevastopol, headed by retired naval Lieutenant Pyotr Schmidt, was directed against the government, while some was undirected.

Kawasaki's theorem, applied to each of the vertices of an arbitrary crease pattern, determines whether the crease pattern is locally flat-foldable, meaning that the part of the crease pattern near the vertex can be flat-folded. However, there exist crease patterns that are locally flat-foldable but that have no global flat folding that works for the whole crease pattern at once. conjectured that global flat-foldability could be tested by checking Kawasaki's theorem at each vertex of a crease pattern, and then also testing bipartiteness of an undirected graph associated with the crease pattern. However, this conjecture was disproven by , who showed that Hull's conditions are not sufficient.

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.

An optimal greedy coloring (left) and Grundy coloring (right) of a crown graph. The numbers indicate the order in which the greedy algorithm colors the vertices. In graph theory, the Grundy number or Grundy chromatic number of an undirected graph is the maximum number of colors that can be used by a greedy coloring strategy that considers the vertices of the graph in sequence and assigns each vertex its first available color, using a vertex ordering chosen to use as many colors as possible. Grundy numbers are named after P. M. Grundy, who studied an analogous concept for directed graphs in 1939.. As cited by .

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.

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.

Chromosome Conformation Capture Technologies Chromosome conformation capture techniques (often abbreviated to 3C technologies or 3C-based methods) are a set of molecular biology methods used to analyze the spatial organization of chromatin in a cell. These methods quantify the number of interactions between genomic loci that are nearby in 3-D space, but may be separated by many nucleotides in the linear genome. Such interactions may result from biological functions, such as promoter-enhancer interactions, or from random polymer looping, where undirected physical motion of chromatin causes loci to collide. Interaction frequencies may be analyzed directly, or they may be converted to distances and used to reconstruct 3-D structures.

Kruskal's algorithm finds a minimum spanning forest of an undirected edge- weighted graph. If the graph is connected, it finds a minimum spanning tree. (A minimum spanning tree of a connected graph is a subset of the edges that forms a tree that includes every vertex, where the sum of the weights of all the edges in the tree is minimized. For a disconnected graph, a minimum spanning forest is composed of a minimum spanning tree for each connected component.) It is a greedy algorithm in graph theory as in each step it adds the next lowest-weight edge that will not form a cycle to the minimum spanning forest.

Not only is music more pleasant to her ears, but the steady beat of contemporary music provides a more constant source of sound to convert. The precise means by which this conversion process works is as yet unknown. Dazzler has been shown to create a "null space" of sound in a certain radius of her person, as a result of "pulling" the sound in her area to her person, to either protect a crowd of people or to supercharge her power reserves.Dazzler: The Movie Marvel Graphic Novel #12 Left undirected, Dazzler's light will radiate from her body in all directions, producing regular flashes of white light.

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.

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.

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.

The simplest topological indices do not recognize double bonds and atom types (C, N, O etc.) and ignore hydrogen atoms ("hydrogen suppressed") and defined for connected undirected molecular graphs only. More sophisticated topological indices also take into account the hybridization state of each of the atoms contained in the molecule. The Hosoya index is the first topological index recognized in chemical graph theory, and it is often referred to as "the" topological index.. Other examples include the Wiener index, Randić's molecular connectivity index, Balaban’s J index, and the TAU descriptors. The extended topochemical atom (ETA) indices have been developed based on refinement of TAU descriptors.

Arctic team maps five islands found by Russian student - BBC News In April 2020, the archipelago was used by the Russian Airborne Forces to perform the world's first HALO paradrop from the lower border of the Arctic stratosphere. The crews of Il-76 aircraft practiced at the northernmost airfield of the country on the island of Franz Josef Land. Not only did the paratroopers endure the partial oxygen of the stratosphere common under the HALO technique; they encountered deep freeze conditions mitigated by military tested oxygen tanks and uniforms. Challenges to the Arctic mission included undirected terrain, in the absence of ground navigation systems.

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.

New Road, Brighton - Shared Space scheme reduced motor traffic by 93%. Shared space schemes extend this principle further by removing the reliance on lane markings altogether, and also removing road signs and signals, allowing all road users to use any part of the road, and giving all road users equal priority and equal responsibility for each other's safety. Experiences where these schemes are in use show that road users, particularly motorists, undirected by signs, kerbs, or road markings, reduce their speed and establish eye contact with other users. Results from the thousands of such implementations worldwide all show casualty reductions and most also show reduced journey times.

A cut of an undirected graph is a partition of the vertices into two nonempty subsets, the sides of the cut. The subset of edges that have one endpoint in each side is called a cut-set. When a cut-set forms a complete bipartite graph, its cut is called a split. Thus, a split can be described as a partition of the vertices of the graph into two subsets and , such that every neighbor of in is adjacent to every neighbor of in .. A cut or split is trivial when one of its two sides has only one vertex in it; every trivial cut is a split.

After high school she attended NYU in the Bronx (uptown campus) for one year, but felt undirected there and decided not to stay. A family friend, Hy Zaret (who wrote the lyrics to "Unchained Melody"), asked her parents if she could sing a demo for him to pitch to Joan Baez, so her mother took her into the city to Aura Recording Studios on 7th Ave. As she was singing in the studio, it was recommended to her mother that Lesley study with Helen Hobbs Jordan, a private music teacher with a studio in the original Steinway Hall. Miller learned music theory, sightreading, and piano, and after years she ventured out to try getting some work.

When two terminal nodes are given, they are typically referred to as the source and the sink. In a directed, weighted flow network, the minimum cut separates the source and sink vertices and minimizes the total weight on the edges that are directed from the source side of the cut to the sink side of the cut. As shown in the max-flow min-cut theorem, the weight of this cut equals the maximum amount of flow that can be sent from the source to the sink in the given network. In a weighted, undirected network, it is possible to calculate the cut that separates a particular pair of vertices from each other and has minimum possible weight.

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.

A bramble of order four in a 3×3 grid graph, consisting of six mutually touching connected subgraphs In graph theory, a bramble for an undirected graph G is a family of connected subgraphs of G that all touch each other: for every pair of disjoint subgraphs, there must exist an edge in G that has one endpoint in each subgraph. The order of a bramble is the smallest size of a hitting set, a set of vertices of G that has a nonempty intersection with each of the subgraphs. Brambles may be used to characterize the treewidth of G.. In this reference, brambles are called "screens" and their order is called "thickness".

Partition of the graph of a rhombic dodecahedron into two linear forests, showing that its linear arboricity is two In graph theory, a branch of mathematics, the linear arboricity of an undirected graph is the smallest number of linear forests its edges can be partitioned into. Here, a linear forest is an acyclic graph with maximum degree two; that is, it is a disjoint union of path graphs. Linear arboricity is a variant of arboricity, the minimum number of forests into which the edges can be partitioned. The linear arboricity of any graph of maximum degree \Delta is known to be at least \lceil\Delta/2\rceil and is conjectured to be at most \lceil(\Delta+1)/2\rceil.

An indifference graph, formed from a set of points on the real line by connecting pairs of points whose distance is at most one In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other.. Indifference graphs are also the intersection graphs of sets of unit intervals, or of properly nested intervals (intervals none of which contains any other one). Based on these two types of interval representations, these graphs are also called unit interval graphs or proper interval graphs; they form a subclass of the interval graphs.

A framework is an undirected graph, embedded into d-dimensional Euclidean space by providing a d-tuple of Cartesian coordinates for each vertex of the graph. From a framework with n vertices and m edges, one can define a matrix with m rows and nd columns, an expanded version of the incidence matrix of the graph called the rigidity matrix. In this matrix, the entry in row e and column (v,i) is zero if v is not an endpoint of edge e. If, on the other hand, edge e has vertices u and v as endpoints, then the value of the entry is the difference between the ith coordinates of v and u.

In any vector space, and more generally in any matroid, a minimum weight basis may be found by a greedy algorithm that considers potential basis elements one at a time, in sorted order by their weights, and that includes an element in the basis when it is linearly independent of the previously chosen basis elements. Testing for linear independence can be done by Gaussian elimination. However, an undirected graph may have an exponentially large set of simple cycles, so it would be computationally infeasible to generate and test all such cycles. provided the first polynomial time algorithm for finding a minimum weight basis, in graphs for which every edge weight is positive.

The usual formulation of the cycle double cover conjecture asks whether every bridgeless undirected graph has a collection of cycles such that each edge of the graph is contained in exactly two of the cycles. The requirement that the graph be bridgeless is an obvious necessary condition for such a set of cycles to exist, because a bridge cannot belong to any cycle. A collection of cycles satisfying the condition of the cycle double cover conjecture is called a cycle double cover. Some graphs such as cycle graphs and bridgeless cactus graphs can only be covered by using the same cycle more than once, so this sort of duplication is allowed in a cycle double cover.

Boston: Houghton Mifflin, 1918. the first textbook published on the subject, in 1918, John Franklin Bobbitt said that curriculum, as an idea, has its roots in the Latin word for race-course, explaining the curriculum as the course of deeds and experiences through which children become the adults they should be to succeed later in life. Furthermore, the curriculum encompasses the entire scope of formative deed and experience occurring in and out of school such as experiences that are unplanned and undirected or those that are intentionally directed for the purposeful formation of adult members of society, not only experiences occurring in school. (cf. image at right.) To Bobbitt, the curriculum is a social engineering arena.

In 1973, in response to the criticism from James Lighthill and ongoing pressure from congress, the U.S. and British Governments stopped funding undirected research into artificial intelligence, and the difficult years that followed would later be known as an "AI winter". Seven years later, a visionary initiative by the Japanese Government inspired governments and industry to provide AI with billions of dollars, but by the late 80s the investors became disillusioned and withdrew funding again. Investment and interest in AI boomed in the first decades of the 21st century, when machine learning was successfully applied to many problems in academia and industry due to new methods, the application of powerful computer hardware, and the collection of immense data sets.

Steiner trees have been extensively studied in the context of weighted graphs. The prototype is, arguably, the Steiner tree problem in graphs. Let G = (V, E) be an undirected graph with non-negative edge weights c and let S ⊆ V be a subset of vertices, called terminals. A Steiner tree is a tree in G that spans S. There are two versions of the problem: in the optimization problem associated with Steiner trees, the task is to find a minimum-weight Steiner tree; in the decision problem the edge weights are integers and the task is to determine whether a Steiner tree exists whose total weight does not exceed a predefined natural number k.

In 1993,Brouwer A. E.; Dejter I. J.; Thomassen C. "Highly symmetric subgraphs of hypercubes", J. Algebraic Combinat. 2, 22-25, 1993 Brouwer, Dejter and Thomassen described an undirected, bipartite graph with 112 vertices and 168 edges, (semi-symmetric, that is edge-transitive but not vertex-transitive, cubic graph with diameter 8, radius 7, chromatic number 2, chromatic index 3, girth 10, with exactly 168 cycles of length 10 and 168 cycles of length 12), known since 2002 as the Ljubljana graph. They also established that the Dejter graph,Klin M.; Lauri J.; Ziv-Av M. "Links between two semisymmetric graphs on 112 vertices through the lens of association schemes", Jour. Symbolic Comput., 47–10, 2012, 1175–1191.

A minor difference between the 2-SAT and initial stable set formulations is that the latter presupposes the choice of a fixed base point from the median graph that corresponds to the empty initial stable set. For a distributive lattice, the corresponding mixed graph has no undirected edges, and the initial stable sets are just the lower sets of the transitive closure of the graph. Equivalently, for a distributive lattice, the implication graph of the 2-satisfiability instance can be partitioned into two connected components, one on the positive variables of the instance and the other on the negative variables; the transitive closure of the positive component is the underlying partial order of the distributive lattice.

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.

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.

Radical thinkers like Jean-Baptiste Lamarck saw a progression of life forms from the simplest creatures striving towards complexity and perfection, a schema accepted by zoologists like Henri de Blainville. The very idea of an ordering of organisms, even if supposedly fixed, laid the basis for the idea of transmutation of species, whether progressive goal-directed orthogenesis or Charles Darwin's undirected theory of evolution. The Chain of Being continued to be part of metaphysics in 19th century education, and the concept was well known. The geologist Charles Lyell used it as a metaphor in his 1851 Elements of Geology description of the geological column, where he used the term "missing links" in relation to missing parts of the continuum.

In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, appeared already in the 13th century, in the work of Ramon Llull.. Such a drawing is sometimes referred to as a mystic rose..

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..

A spanning tree (blue heavy edges) of a grid graph In the mathematical field of graph theory, a spanning tree T of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G, with a minimum possible number of edges. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree (but see spanning forests below). If all of the edges of G are also edges of a spanning tree T of G, then G is a tree and is identical to T (that is, a tree has a unique spanning tree and it is itself).

In this graph, the red triangle formed by vertices 1, 2, and 5 is a peripheral cycle: the four remaining edges form a single bridge. However, pentagon 1–2–3–4–5 is not peripheral, as the two remaining edges form two separate bridges. In graph theory, a peripheral cycle (or peripheral circuit) in an undirected graph is, intuitively, a cycle that does not separate any part of the graph from any other part. Peripheral cycles (or, as they were initially called, peripheral polygons, because Tutte called cycles "polygons") were first studied by , and play important roles in the characterization of planar graphs and in generating the cycle spaces of nonplanar graphs..

The four types of edges defined by a spanning tree A convenient description of a depth-first search of a graph is in terms of a spanning tree of the vertices reached during the search. Based on this spanning tree, the edges of the original graph can be divided into three classes: forward edges, which point from a node of the tree to one of its descendants, back edges, which point from a node to one of its ancestors, and cross edges, which do neither. Sometimes tree edges, edges which belong to the spanning tree itself, are classified separately from forward edges. If the original graph is undirected then all of its edges are tree edges or back edges.

The Bondy–Chvátal theorem states that a graph is Hamiltonian if and only if its closure is Hamiltonian; since the complete graph is Hamiltonian, Ore's theorem is an immediate consequence. found a version of Ore's theorem that applies to directed graphs. Suppose a digraph G has the property that, for every two vertices u and v, either there is an edge from u to v or the outdegree of u plus the indegree of v equals or exceeds the number of vertices in G. Then, according to Woodall's theorem, G contains a directed Hamiltonian cycle. Ore's theorem may be obtained from Woodall by replacing every edge in a given undirected graph by a pair of directed edges.

Another equivalent formulation of the union-closed sets conjecture uses graph theory. In an undirected graph, an independent set is a set of vertices no two of which are adjacent to each other; an independent set is maximal if it is not a subset of a larger independent set. In any graph, the "heavy" vertices that appear in more than half of the maximal independent sets must themselves form an independent set, so there always exists at least one non- heavy vertex, a vertex that appears in at most half of the maximal independent sets. The graph formulation of the union-closed sets conjecture states that every finite non-empty graph contains two adjacent non-heavy vertices.

An unrooted binary tree is a connected undirected graph with no cycles in which each non-leaf node has exactly three neighbors. A branch-decomposition may be represented by an unrooted binary tree T, together with a bijection between the leaves of T and the edges of the given graph G = (V,E). If e is any edge of the tree T, then removing e from T partitions it into two subtrees T1 and T2. This partition of T into subtrees induces a partition of the edges associated with the leaves of T into two subgraphs G1 and G2 of G. This partition of G into two subgraphs is called an e-separation.

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.

In some applications, such cycles are undesirable, and we wish to eliminate them and obtain a directed acyclic graph (DAG). One way to do this is simply to drop edges from the graph to break the cycles. Closely related are the feedback vertex set, which is a set of vertices containing at least one vertex from every cycle in the directed graph, and the minimum spanning tree, which is the undirected variant of the feedback arc set problem. A minimal feedback arc set (one that can not be reduced in size by removing any edges) has the additional property that, if the edges in it are reversed rather than removed, then the graph remains acyclic.

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.

We usually consider algorithms in L, the class of problems requiring O(log n) additional space, to be in-place. This class is more in line with the practical definition, as it allows numbers of size n as pointers or indices. This expanded definition still excludes quicksort, however, because of its recursive calls. Identifying the in-place algorithms with L has some interesting implications; for example, it means that there is a (rather complex) in-place algorithm to determine whether a path exists between two nodes in an undirected graph, a problem that requires O(n) extra space using typical algorithms such as depth-first search (a visited bit for each node).

In computational complexity theory, a planted clique or hidden clique in an undirected graph is a clique formed from another graph by selecting a subset of vertices and adding edges between each pair of vertices in the subset. The planted clique problem is the algorithmic problem of distinguishing random graphs from graphs that have a planted clique. This is a variation of the clique problem; it may be solved in quasi-polynomial time but is conjectured not to be solvable in polynomial time for intermediate values of the clique size. The conjecture that no polynomial time solution exists is called the planted clique conjecture; it has been used as a computational hardness assumption.

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.

In the weighted 2-satisfiability problem (W2SAT), the input is an n-variable 2SAT instance and an integer , and the problem is to decide whether there exists a satisfying assignment in which at most of the variables are true. The W2SAT problem includes as a special case the vertex cover problem, of finding a set of vertices that together touch all the edges of a given undirected graph. For any given instance of the vertex cover problem, one can construct an equivalent W2SAT problem with a variable for each vertex of a graph. Each edge of the graph may be represented by a 2SAT clause that can be satisfied only by including either or among the true variables of the solution.

A graph with n nodes can contain at most n-1 bridges, since adding additional edges must create a cycle. The graphs with exactly n-1 bridges are exactly the trees, and the graphs in which every edge is a bridge are exactly the forests. In every undirected graph, there is an equivalence relation on the vertices according to which two vertices are related to each other whenever there are two edge-disjoint paths connecting them. (Every vertex is related to itself via two length-zero paths, which are identical but nevertheless edge-disjoint.) The equivalence classes of this relation are called 2-edge-connected components, and the bridges of the graph are exactly the edges whose endpoints belong to different components.

First steps toward determining the computational complexity were undertaken in proving that the problem is in larger complexity classes, which contain the class P. By using normal surfaces to describe the Seifert surfaces of a given knot, showed that the unknotting problem is in the complexity class NP. claimed the weaker result that unknotting is in AM ∩ co- AM; however, later they retracted this claim.Mentioned as a "personal communication" in reference [15] of . In 2011, Greg Kuperberg proved that (assuming the generalized Riemann hypothesis) the unknotting problem is in co- NP, and in 2016, Marc Lackenby provided an unconditional proof of co-NP membership. The unknotting problem has the same computational complexity as testing whether an embedding of an undirected graph in Euclidean space is linkless.

The symmetric difference of two cycles is an Eulerian subgraph In graph theory, a branch of mathematics, a cycle basis of an undirected graph is a set of simple cycles that forms a basis of the cycle space of the graph. That is, it is a minimal set of cycles that allows every even-degree subgraph to be expressed as a symmetric difference of basis cycles. A fundamental cycle basis may be formed from any spanning tree or spanning forest of the given graph, by selecting the cycles formed by the combination of a path in the tree and a single edge outside the tree. Alternatively, if the edges of the graph have positive weights, the minimum weight cycle basis may be constructed in polynomial time.

In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non- adjacent vertices. It is also called the Seidel matrix or--its original name-- the (−1,1,0)-adjacency matrix. It can be interpreted as the result of subtracting the adjacency matrix of G from the adjacency matrix of the complement of G. The multiset of eigenvalues of this matrix is called the Seidel spectrum. The Seidel matrix was introduced by J. H. van Lint and J. J. Seidel in 1966 and extensively exploited by Seidel and coauthors.

Mac Lane's planarity criterion, named after Saunders Mac Lane, characterizes planar graphs in terms of their cycle spaces and cycle bases. It states that a finite undirected graph is planar if and only if the graph has a cycle basis in which each edge of the graph participates in at most two basis cycles. In a planar graph, a cycle basis formed by the set of bounded faces of an embedding necessarily has this property: each edge participates only in the basis cycles for the two faces it separates. Conversely, if a cycle basis has at most two cycles per edge, then its cycles can be used as the set of bounded faces of a planar embedding of its graph..

In graph theory, a cop-win graph is an undirected graph on which the pursuer (cop) can always win a pursuit-evasion game in which he chases a robber, the players alternately moving along an edge of a graph or staying put, until the cop lands on the robber's vertex.. Finite cop-win graphs are also called dismantlable graphs or constructible graphs, because they can be dismantled by repeatedly removing a dominated vertex (one whose closed neighborhood is a subset of another vertex's neighborhood) or constructed by repeatedly adding such a vertex. The cop-win graphs can be recognized in polynomial time by a greedy algorithm that constructs a dismantling order. They include the chordal graphs, and the graphs that contain a universal vertex.

The median of three vertices in a median graph In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices a, b, and c have a unique median: a vertex m(a,b,c) that belongs to shortest paths between each pair of a, b, and c. The concept of median graphs has long been studied, for instance by or (more explicitly) by , but the first paper to call them "median graphs" appears to be . As Chung, Graham, and Saks write, "median graphs arise naturally in the study of ordered sets and discrete distributive lattices, and have an extensive literature".. In phylogenetics, the Buneman graph representing all maximum parsimony evolutionary trees is a median graph.; ; .

In graph theory, a Trémaux tree of an undirected graph G is a spanning tree of G, rooted at one of its vertices, with the property that every two adjacent vertices in G are related to each other as an ancestor and descendant in the tree. All depth-first search trees and all Hamiltonian paths are Trémaux trees. Trémaux trees are named after Charles Pierre Trémaux, a 19th-century French author who used a form of depth-first search as a strategy for solving mazes... They have also been called normal spanning trees, especially in the context of infinite graphs. In finite graphs, although depth-first search itself is inherently sequential, Trémaux trees can be constructed by a randomized parallel algorithm in the complexity class RNC.

Every finite connected undirected graph has at least one Trémaux tree. One can construct such a tree by performing a depth-first search and connecting each vertex (other than the starting vertex of the search) to the earlier vertex from which it was discovered. The tree constructed in this way is known as a depth-first search tree. If uv is an arbitrary edge in the graph, and u is the earlier of the two vertices to be reached by the search, then v must belong to the subtree descending from u in the depth-first search tree, because the search will necessarily discover v while it is exploring this subtree, either from one of the other vertices in the subtree or, failing that, from u directly.

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.

The proof goes as follows: First, the polygon is triangulated (without adding extra vertices). The vertices of the resulting triangulation graph may be 3-colored.To prove 3-colorability of polygon triangulations, we observe that the weak dual graph to the triangulation (the undirected graph having one vertex per triangle and one edge per pair of adjacent triangles) is a tree, since any cycle in the dual graph would form the boundary of a hole in the polygon, contrary to the assumption that it has no holes. Whenever there is more than one triangle, the dual graph (like any tree) must have a vertex with only one neighbor, corresponding to a triangle that is adjacent to other triangles along only one of its sides.

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.

In computational complexity theory, PPA is a complexity class, standing for "Polynomial Parity Argument" (on a graph). Introduced by Christos Papadimitriou in 1994 (page 528), PPA is a subclass of TFNP. It is a class of search problems that can be shown to be total by an application of the handshaking lemma: any undirected graph that has a vertex whose degree is an odd number must have some other vertex whose degree is an odd number. This observation means that if we are given a graph and an odd-degree vertex, and we are asked to find some other odd-degree vertex, then we are searching for something that is guaranteed to exist (so, we have a total search problem).

An example showing how the FKT algorithm finds a Pfaffian orientation. # Compute a planar embedding of G. # Compute a spanning tree T1 of the input graph G. # Give an arbitrary orientation to each edge in G that is also in T1. # Use the planar embedding to create an (undirected) graph T2 with the same vertex set as the dual graph of G. # Create an edge in T2 between two vertices if their corresponding faces in G share an edge in G that is not in T1. (Note that T2 is a tree.) # For each leaf v in T2 (that is not also the root): ## Let e be the lone edge of G in the face corresponding to v that does not yet have an orientation.

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.

Consider the case where Y is the graph with vertex set {1,2,3} and undirected edges {1,2}, {1,3} and {2,3} (a triangle or 3-circle) with vertex states from K = {0,1}. For vertex functions use the symmetric, boolean function nor : K3 → K defined by nor(x,y,z) = (1+x)(1+y)(1+z) with boolean arithmetic. Thus, the only case in which the function nor returns the value 1 is when all the arguments are 0. Pick w = (1,2,3) as update sequence. Starting from the initial system state (0,0,0) at time t = 0 one computes the state of vertex 1 at time t=1 as nor(0,0,0) = 1. The state of vertex 2 at time t=1 is nor(1,0,0) = 0.

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.

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.

For the purposes of defining the crossing number, a drawing of an undirected graph is a mapping from the vertices of the graph to disjoint points in the plane, and from the edges of the graph to curves connecting their two endpoints. No vertex should be mapped onto an edge that it is not an endpoint of, and whenever two edges have curves that intersect (other than at a shared endpoint) their intersections should form a finite set of proper crossings, where the two curves are transverse. A crossing is counted separately for each of these crossing points, for each pair of edges that cross. The crossing number of a graph is then the minimum, over all such drawings, of the number of crossings in a drawing.

The existence of well-balanced orientations, together with Menger's theorem, immediately implies Robbins' theorem: by Menger's theorem, a 2-edge-connected graph has at least two edge- disjoint paths between every pair of vertices, from which it follows that any well-balanced orientation must be strongly connected. More generally this result implies that every -edge-connected undirected graph can be oriented to form a -edge-connected directed graph. A totally cyclic orientation of a graph is an orientation in which each edge belongs to a directed cycle. For connected graphs, this is the same thing as a strong orientation, but totally cyclic orientations may also be defined for disconnected graphs, and are the orientations in which each connected component of becomes strongly connected.

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.

Let G = ((VG, EG), c) be an undirected graph with c(u,v) being the capacity of the edge (u,v) respectively. : Denote the minimum capacity of an s-t cut by λst for each s, t ∈ VG. : Let T = (VT,ET) be a tree with VT = VG, denote the set of edges in an s-t path by Pst for each s,t ∈ VT. Then T is said to be a Gomory–Hu tree of G if : λst = mine∈Pst c(Se, Te) for all s, t ∈ VG, where # Se and Te are the two connected components of T∖{e} in the sense that (Se, Te) form a s-t cut in G, and # c(Se, Te) is the capacity of the cut in G.

The matroid partitioning problem is a problem arising in the mathematical study of matroids and in the design and analysis of algorithms, in which the goal is to partition the elements of a matroid into as few independent sets as possible. An example is the problem of computing the arboricity of an undirected graph, the minimum number of forests needed to cover all of its edges. Matroid partitioning may be solved in polynomial time, given an independence oracle for the matroid. It may be generalized to show that a matroid sum is itself a matroid, to provide an algorithm for computing ranks and independent sets in matroid sums, and to compute the largest common independent set in the intersection of two given matroids..

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.

In network theory, the Wiener connector is a means of maximizing efficiency in connecting specified "query vertices" in a network. Given a connected, undirected graph and a set of query vertices in a graph, the minimum Wiener connector is an induced subgraph that connects the query vertices and minimizes the sum of shortest path distances among all pairs of vertices in the subgraph. In combinatorial optimization, the minimum Wiener connector problem is the problem of finding the minimum Wiener connector. It can be thought of as a version of the classic Steiner tree problem (one of Karp's 21 NP-complete problems), where instead of minimizing the size of the tree, the objective is to minimize the distances in the subgraph.

Biologists and Earth scientists usually view the factors that stabilize the characteristics of a period as an undirected emergent property or entelechy of the system; as each individual species pursues its own self-interest, for example, their combined actions may have counterbalancing effects on environmental change. Opponents of this view sometimes reference examples of events that resulted in dramatic change rather than stable equilibrium, such as the conversion of the Earth's atmosphere from a reducing environment to an oxygen-rich one at the end of the Archaean and the beginning of the Proterozoic periods. Less accepted versions of the hypothesis claim that changes in the biosphere are brought about through the coordination of living organisms and maintain those conditions through homeostasis.

Both checking whether a 2-coloring is valid and checking whether a given odd-length sequence of vertices is a cycle may be performed more simply than testing bipartiteness. Analogously, it is possible to test whether a given directed graph is acyclic by a certifying algorithm that outputs either a topological order or a directed cycle. It is possible to test whether an undirected graph is a chordal graph by a certifying algorithm that outputs either an elimination ordering (an ordering of all vertices such that, for every vertex, the neighbors that are later in the ordering form a clique) or a chordless cycle. And it is possible to test whether a graph is planar by a certifying algorithm that outputs either a planar embedding or a Kuratowski subgraph.

A Markov network or MRF is similar to a Bayesian network in its representation of dependencies; the differences being that Bayesian networks are directed and acyclic, whereas Markov networks are undirected and may be cyclic. Thus, a Markov network can represent certain dependencies that a Bayesian network cannot (such as cyclic dependencies ); on the other hand, it can't represent certain dependencies that a Bayesian network can (such as induced dependencies ). The underlying graph of a Markov random field may be finite or infinite. When the joint probability density of the random variables is strictly positive, it is also referred to as a Gibbs random field, because, according to the Hammersley–Clifford theorem, it can then be represented by a Gibbs measure for an appropriate (locally defined) energy function.

A median graph is an undirected graph in which for every three vertices x, y, and z there is a unique vertex \langle x,y,z \rangle that belongs to shortest paths between any two of x, y, and z. If this is the case, then the operation \langle x,y,z \rangle defines a median algebra having the vertices of the graph as its elements. Conversely, in any median algebra, one may define an interval [x, z] to be the set of elements y such that \langle x,y,z \rangle = y. One may define a graph from a median algebra by creating a vertex for each algebra element and an edge for each pair (x, z) such that the interval [x, z] contains no other elements.

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.

The face cycles of these embeddings generate a proper subset of all Eulerian subgraphs. The homology group H_2(S,\Z_2) of the given surface S characterizes the Eulerian subgraphs that cannot be represented as the boundary of a set of faces. Mac Lane's planarity criterion uses this idea to characterize the planar graphs in terms of the cycle bases: a finite undirected graph is planar if and only if it has a sparse cycle basis or 2-basis, a basis in which each edge of the graph participates in at most two basis cycles. In a planar graph, the cycle basis formed by the set of bounded faces is necessarily sparse, and conversely, a sparse cycle basis of any graph necessarily forms the set of bounded faces of a planar embedding of its graph..

The identity element of an abelian sandpile model A chip-firing game, in its most basic form, is a process on an undirected graph, with each vertex of the graph containing some number of chips. At each step, a vertex with more chips than incident edges is selected, and one of its chips is sent to each of its neighbors. If a single vertex is designated as a "black hole", meaning that chips sent to it vanish, then the result of the process is the same no matter what order the other vertices are selected. The stable states of this process are the ones in which no vertex has enough chips to be selected; two stable states can be added by combining their chips and then stabilizing the result.

The Hammersley–Clifford theorem is a result in probability theory, mathematical statistics and statistical mechanics that gives necessary and sufficient conditions under which a strictly positive probability distribution (of events in a probability space) can be represented as events generated by a Markov network (also known as a Markov random field). It is the fundamental theorem of random fields. It states that a probability distribution that has a strictly positive mass or density satisfies one of the Markov properties with respect to an undirected graph G if and only if it is a Gibbs random field, that is, its density can be factorized over the cliques (or complete subgraphs) of the graph. The relationship between Markov and Gibbs random fields was initiated by Roland Dobrushin and Frank Spitzer in the context of statistical mechanics.

Then, a Hamiltonian path exists if and only if there is a set of n − 1 elements in the intersection of three matroids on the edge set of the graph: two partition matroids ensuring that the in-degree and out-degree of the selected edge set are both at most one, and the graphic matroid of the undirected graph formed by forgetting the edge orientations in G, ensuring that the selected edge set has no cycles . Another computational problem on matroids, the matroid parity problem, was formulated by as a common generalization of matroid intersection and non-bipartite graph matching. However, although it can be solved in polynomial time for linear matroids, it is NP-hard for other matroids, and requires exponential time in the matroid oracle model .

Rather, the system is intended to counteract incidental and undirected yawing motions, which can be characterised as skids or slips. On a single-engine aircraft, the system is particularly useful at addressing the tendency to 'fishtail', smoothing out the left-right movements of the vertical stabilizer, increasing ride comfort. It is also particularly useful on swept wing aircraft, particularly those using a T-tail arrangement; without an active yaw damper system, these types of aircraft are susceptible to the Dutch roll phenomenon, where yawing motions can result in repetitive corkscrew-like oscillations that could potentially escalate to excessive levels if not effectively counteracted. The yaw damper is typically disengaged at ground level and turned on shortly after takeoff; an active yaw damper during the takeoff run could potentially mask serious issues such as engine failure.

An unrooted binary tree T may be transformed into a full rooted binary tree (that is, a rooted tree in which each non-leaf node has exactly two children) by choosing a root edge e of T, placing a new root node in the middle of e, and directing every edge of the resulting subdivided tree away from the root node. Conversely, any full rooted binary tree may be transformed into an unrooted binary tree by removing the root node, replacing the path between its two children by a single undirected edge, and suppressing the orientation of the remaining edges in the graph. For this reason, there are exactly 2n −3 times as many full rooted binary trees with n leaves as there are unrooted binary trees with n leaves.

The input to the algorithm is an undirected graph with vertex set , edge set , and (optionally) numerical weights on the edges in . The goal of the algorithm is to partition into two disjoint subsets and of equal (or nearly equal) size, in a way that minimizes the sum of the weights of the subset of edges that cross from to . If the graph is unweighted, then instead the goal is to minimize the number of crossing edges; this is equivalent to assigning weight one to each edge. The algorithm maintains and improves a partition, in each pass using a greedy algorithm to pair up vertices of with vertices of , so that moving the paired vertices from one side of the partition to the other will improve the partition.

Tietze's subdivision of a Möbius strip into six mutually-adjacent regions. The vertices and edges of the subdivision form an embedding of Tietze's graph onto the strip. In the mathematical field of graph theory, Tietze's graph is an undirected cubic graph with 12 vertices and 18 edges. It is named after Heinrich Franz Friedrich Tietze, who showed in 1910 that the Möbius strip can be subdivided into six regions that all touch each other – three along the boundary of the strip and three along its center line – and therefore that graphs that are embedded onto the Möbius strip may require six colors.. The boundary segments of the regions of Tietze's subdivision (including the segments along the boundary of the Möbius strip itself) form an embedding of Tietze's graph.

In this problem, one must choose time slots for the edges of a wireless communications network so that each node of the network can communicate with each neighboring node without interference. Using a strong edge coloring (and using two time slots for each edge color, one for each direction) would solve the problem but might use more time slots than necessary. Instead, they seek a coloring of the directed graph formed by doubling each undirected edge of the network, with the property that each directed edge has a different color from the edges that go out from and from the neighbors of . They propose a heuristic for this problem based on a distributed algorithm for -edge-coloring together with a postprocessing phase that reschedules edges that might interfere with each other.

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.

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.

The shortest-path graph with t=2 In mathematics and geographic information science, a shortest-path graph is an undirected graph defined from a set of points in the Euclidean plane. The shortest-path graph is proposed with the idea of inferring edges between a point set such that the shortest path taken over the inferred edges will roughly align with the shortest path taken over the imprecise region represented by the point set. The edge set of the shortest-path graph varies based on a single parameter t ≥ 1. When the weight of an edge is defined as its Euclidean length raised to the power of the parameter t ≥ 1, the edge is present in the shortest-path graph if and only if it is the least weight path between its endpoints.

A graph that requires four colors in any coloring, and four connected subgraphs that, when contracted, form a complete graph, illustrating the case k = 4 of Hadwiger's conjecture In graph theory, the Hadwiger conjecture states that if G is loopless and has no K_t minor then its chromatic number satisfies \chi(G) < t. It is known to be true for 1 \leq t \leq 6. The conjecture is a generalization of the four-color theorem and is considered to be one of the most important and challenging open problems in the field. In more detail, if all proper colorings of an undirected graph G use k or more colors, then one can find k disjoint connected subgraphs of G such that each subgraph is connected by an edge to each other subgraph.

An Apollonian network, the graph of a stacked polyhedron The undirected graph formed by the vertices and edges of a stacked polytope in d dimensions is a (d + 1)-tree. More precisely, the graphs of stacked polytopes are exactly the (d + 1)-trees in which every d-vertex clique (complete subgraph) is contained in at most two (d + 1)-vertex cliques.. See in particular p. 420. For instance, the graphs of three-dimensional stacked polyhedra are exactly the Apollonian networks, the graphs formed from a triangle by repeatedly subdividing a triangular face of the graph into three smaller triangles. One reason for the significance of stacked polytopes is that, among all d-dimensional simplicial polytopes with a given number of vertices, the stacked polytopes have the fewest possible higher-dimensional faces.

Mirsky was inspired by Dilworth's theorem, stating that, for every partially ordered set, the maximum size of an antichain equals the minimum number of chains in a partition of the set into chains. For sets of order dimension two, the two theorems coincide (a chain in the majorization ordering of points in general position in the plane is an antichain in the set of points formed by a 90° rotation from the original set, and vice versa) but for more general partial orders the two theorems differ, and (as Mirsky observes) Dilworth's theorem is more difficult to prove. Mirsky's theorem and Dilworth's theorem are also related to each other through the theory of perfect graphs. An undirected graph is perfect if, in every induced subgraph, the chromatic number equals the size of the largest clique.

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.

Any undirected graph G may be represented as an intersection graph: for each vertex vi of G, form a set Si consisting of the edges incident to vi; then two such sets have a nonempty intersection if and only if the corresponding vertices share an edge. provide a construction that is more efficient (which is to say requires a smaller total number of elements in all of the sets Si combined) in which the total number of set elements is at most n2/4 where n is the number of vertices in the graph. They credit the observation that all graphs are intersection graphs to , but say to see also . The intersection number of a graph is the minimum total number of elements in any intersection representation of the graph.

Channel routing example Channel routing is the problem of routing of a set of nets N which have fixed terminals on two opposite sides of a rectangle ("channel"). In this context, the horizontal constraint graph is the undirected graph with vertex set N and two nets are connected by an edge if and only if horizontal segments of the routing must overlap. In the given example, only nets 5 and 6 do not have a horizontal constraint between them. The vertical constraint graph is the directed graph with vertex set N and two nets are connected by an edge if and only if there are two pins from different nets on the same vertical line and the edge is directed from the net with pin on the upper edge of the channel.

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 .

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.

Binary search has been generalized to work on certain types of graphs, where the target value is stored in a vertex instead of an array element. Binary search trees are one such generalization--when a vertex (node) in the tree is queried, the algorithm either learns that the vertex is the target, or otherwise which subtree the target would be located in. However, this can be further generalized as follows: given an undirected, positively weighted graph and a target vertex, the algorithm learns upon querying a vertex that it is equal to the target, or it is given an incident edge that is on the shortest path from the queried vertex to the target. The standard binary search algorithm is simply the case where the graph is a path.

If G is an undirected graph, and X is a set of vertices, then an X-flap is a nonempty connected component of the subgraph of G formed by deleting X. A haven of order k in G is a function β that assigns an X-flap β(X) to every set X of fewer than k vertices. This function must also satisfy additional constraints which are given differently by different authors. The number k is called the order of the haven.. In the original definition of Seymour and Thomas,. a haven is required to satisfy the property that every two flaps β(X) and β(Y) must touch each other: either they share a common vertex or there exists an edge with one endpoint in each flap.

However, the nature of their expressions are significantly different from the one studied by Hassan et al.. Yet another closely related model is the Growing Network with Redirection (GNR) model presented by Gabel, Krapivsky and Redner where at each time step a new node either attaches to a randomly chosen target node with probability 1-r, or to the parent of the target with probability r=1. The GNR model with r=1 may appear similar to the MDA model. However, unlike the GNR model, the MDA model is for undirected networks, and that the new link can connect with any neighbor of the mediator-parent or not. One more difference is that, in the MDA model new node may join the existing network with m edges and in the GNR model it is considered m=1 case only.

Thus, the 1-forests are exactly the pseudoforests in which every component is a 1-tree. :The spanning pseudoforests of an undirected graph G are the pseudoforest subgraphs of G that have all the vertices of G. Such a pseudoforest need not have any edges, since for example the subgraph that has all the vertices of G and no edges is a pseudoforest (whose components are trees consisting of a single vertex). :The maximal pseudoforests of G are the pseudoforest subgraphs of G that are not contained within any larger pseudoforest of G. A maximal pseudoforest of G is always a spanning pseudoforest, but not conversely. If G has no connected components that are trees, then its maximal pseudoforests are 1-forests, but if G does have a tree component, its maximal pseudoforests are not 1-forests.

The term "message in a bottle" has been applied to techniques of communication that do not literally involve a bottle or a water-based method of conveyance, such as the Pioneer plaque (1972, 1973), the Voyager Golden Record (1977), and even radio-borne messages (see Cosmic Call, Teen Age Message, A Message from Earth), all directed into space. Balloon mail involves sending undirected messages through the air rather than into bodies of water. For example, during the Prussian siege of Paris in 1870, about 2.5 million letters were sent by hot air balloon, the only way Parisians' letters could reach the rest of France. Stationary time capsules have been termed "messages in a bottle", such as a 1935 message in a lemonade bottle correctly portending difficult times, which was found in 2016 by masons restoring damaged Portland stone at Southampton Guildhall.

Neo4j's database supports undocumented graph-wide properties, Tinkerpop has graph values which play the same role, and also supports "metaproperties" or properties on properties. Oracle's PGQL supports zero to many labels on nodes and on edges, whereas SQL/PGQ supports one to many labels for each kind of element. The GQL project will define a standard data model, which is likely to be the superset of these variants, and at least the first version of GQL is likely to permit vendors to decide on the cardinalities of labels in each implementation, as does SQL/PGQ, and to choose whether to support undirected relationships. Additional aspects of the ERM or UML models (like generalization or subtyping, or entity or relationship cardinalities) may be captured by GQL schemas or types that describe possible instances of the general data model.

The original application of chordal completion described in Computers and Intractability involves Gaussian elimination for sparse matrices. During the process of Gaussian elimination, one wishes to minimize fill-in, coefficients of the matrix that were initially zero but later become nonzero, because the need to calculate the values of these coefficients slows down the algorithm. The pattern of nonzeros in a sparse symmetric matrix can be described by an undirected graph (having the matrix as its adjacency matrix); the pattern of nonzeros in the filled-in matrix is always a chordal graph, any minimal chordal completion corresponds to a fill-in pattern in this way. If a chordal completion of a graph is given, a sequence of steps in which to perform Gaussian elimination to achieve this fill-in pattern can be found by computing an elimination ordering of the resulting chordal graph.

In theoretical computer science, the Aanderaa–Karp–Rosenberg conjecture (also known as the Aanderaa–Rosenberg conjecture or the evasiveness conjecture) is a group of related conjectures about the number of questions of the form "Is there an edge between vertex u and vertex v?" that have to be answered to determine whether or not an undirected graph has a particular property such as planarity or bipartiteness. They are named after Stål Aanderaa, Richard M. Karp, and Arnold L. Rosenberg. According to the conjecture, for a wide class of properties, no algorithm can guarantee that it will be able to skip any questions: any algorithm for determining whether the graph has the property, no matter how clever, might need to examine every pair of vertices before it can give its answer. A property satisfying this conjecture is called evasive.

However, every planar graph has an arc diagram in which each edge is drawn as a biarc with at most two semicircles. More strongly, every st-planar directed graph (a planar directed acyclic graph with a single source and a single sink, both on the outer face) has an arc diagram in which every edge forms a monotonic curve, with these curves all consistently oriented from one end of the vertex line towards the other. For undirected planar graphs, one way to construct an arc diagram with at most two semicircles per edge is to subdivide the graph and add extra edges so that the resulting graph has a Hamiltonian cycle (and so that each edge is subdivided at most once), and to use the ordering of the vertices on the Hamiltonian cycle as the ordering along the line.

The family of all cut sets of an undirected graph is known as the cut space of the graph. It forms a vector space over the two- element finite field of arithmetic modulo two, with the symmetric difference of two cut sets as the vector addition operation, and is the orthogonal complement of the cycle space... If the edges of the graph are given positive weights, the minimum weight basis of the cut space can be described by a tree on the same vertex set as the graph, called the Gomory–Hu tree.. Each edge of this tree is associated with a bond in the original graph, and the minimum cut between two nodes s and t is the minimum weight bond among the ones associated with the path from s to t in the tree.

Insertions succeed in expected constant time, even considering the possibility of having to rebuild the table, as long as the number of keys is kept below half of the capacity of the hash table, i.e., the load factor is below 50%. One method of proving this uses the theory of random graphs: one may form an undirected graph called the "cuckoo graph" that has a vertex for each hash table location, and an edge for each hashed value, with the endpoints of the edge being the two possible locations of the value. Then, the greedy insertion algorithm for adding a set of values to a cuckoo hash table succeeds if and only if the cuckoo graph for this set of values is a pseudoforest, a graph with at most one cycle in each of its connected components.

The degeneracy of an undirected graph is the smallest number such that every non-empty subgraph of has at least one vertex of degree at most . If one repeatedly removes a minimum-degree vertex from until no vertices are left, then the largest of the degrees of the vertices at the time of their removal will be exactly , and this method of repeated removal can be used to compute the degeneracy of any graph in linear time. Greedy coloring the vertices in the reverse of this removal ordering will automatically produce a coloring with at most colors, and for some graphs (such as complete graphs and odd-length cycle graphs) this number of colors is optimal. For colorings with colors, it may not be possible to move from one coloring to another by changing the color of one vertex at a time.

The incidence poset of any undirected graph G has the vertices and edges of G as its elements; in this poset, x ≤ y if either x = y or x is a vertex, y is an edge, and x is an endpoint of y. Certain kinds of graphs may be characterized by the order dimensions of their incidence posets: a graph is a path graph if and only if the order dimension of its incidence poset is at most two, and according to Schnyder's theorem it is a planar graph if and only if the order dimension of its incidence poset is at most three . For a complete graph on n vertices, the order dimension of the incidence poset is \Theta(\log\log n) . It follows that all simple n-vertex graphs have incidence posets with order dimension O(\log\log n).

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.

If edges (to child nodes) are thought of as references, then a tree is a special case of a digraph, and the tree data structure can be generalized to represent directed graphs by removing the constraints that a node may have at most one parent, and that no cycles are allowed. Edges are still abstractly considered as pairs of nodes, however, the terms and are usually replaced by different terminology (for example, and ). Different implementation strategies exist: a digraph can be represented by the same local data structure as a tree (node with value and list of children), assuming that "list of children" is a list of references, or globally by such structures as adjacency lists. In graph theory, a tree is a connected acyclic graph; unless stated otherwise, in graph theory trees and graphs are assumed undirected.

Inclusive Fitness theory has often been interpreted to mean that social behavior per se is a goal of evolution, and also that genes (or individual organisms) are selected to find ways of actively distinguishing the identity of close genetic relatives ‘in order to’ engage in social behaviors with them. The apparent rationale for this common mis-interpretation is that organisms would thereby benefit the “Inclusive Fitness of the individuals (and genes) involved”. This approach overlooks the point that evolution is not a teleological process, but a passive, consequential and undirected biological process, where environmental variations and drift effects are present alongside random gene mutations and natural selection. Inclusive fitness theory takes the form of an ultimate explanation, specifically a criterion (br>c), for the evolution of social behaviors, not a proximate mechanism governing the expression of social behaviors.

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.

In graph theory, the bipartite double cover of an undirected graph G is a bipartite covering graph of G, with twice as many vertices as G. It can be constructed as the tensor product of graphs, G × K2. It is also called the Kronecker double cover, canonical double cover or simply the bipartite double of G. It should not be confused with a cycle double cover of a graph, a family of cycles that includes each edge twice. In a connected graph that is not bipartite, only one double cover is bipartite, but when the graph is bipartite or disconnected there may be more than one. For this reason, Tomaž Pisanski has argued that the name "bipartite double cover" should be deprecated in favor of the "canonical double cover" or "Kronecker cover" names, which are unambiguous.

200px In computational geometry, the Yao graph, named after Andrew Yao, is a kind of geometric spanner, a weighted undirected graph connecting a set of geometric points with the property that, for every pair of points in the graph, their shortest path has a length that is within a constant factor of their Euclidean distance. The basic idea underlying the two-dimensional Yao graph is to surround each of the given points by equally spaced rays, partitioning the plane into sectors with equal angles, and to connect each point to its nearest neighbor in each of these sectors. Associated with a Yao graph is an integer parameter which is the number of rays and sectors described above; larger values of produce closer approximations to the Euclidean distance. The stretch factor is at most 1/(\cos \theta - \sin \theta), where \theta is the angle of the sectors.

This graph has circuit rank because it can be made into a tree by removing two edges, for instance the edges 1–2 and 2–3, but removing any one edge leaves a cycle in the graph. In graph theory, a branch of mathematics, the circuit rank, cyclomatic number, cycle rank, or nullity of an undirected graph is the minimum number of edges that must be removed from the graph to break all its cycles, making it into a tree or forest. It is equal to the number of independent cycles in the graph (the size of a cycle basis). Unlike the corresponding feedback arc set problem for directed graphs, the circuit rank is easily computed using the formula :r = m - n + c, where is the number of edges in the given graph, is the number of vertices, and is the number of connected components. .

Several combinatorial structures listed by may be shown to be even in number by relating them to the odd vertices in an appropriate "exchange graph". For instance, as C. A. B. Smith proved, in any cubic graph G there must be an even number of Hamiltonian cycles through any fixed edge uv; used a proof based on the handshaking lemma to extend this result to graphs G in which all vertices have odd degree. Thomason defines an exchange graph H, the vertices of which are in one-to-one correspondence with the Hamiltonian paths beginning at u and continuing through v. Two such paths p1 and p2 are connected by an edge in H if one may obtain p2 by adding a new edge to the end of p1 and removing another edge from the middle of p1; this is a symmetric relation, so H is an undirected graph.

An abstract polytope is a partially ordered set P (whose elements are called faces) with properties modeling those of the inclusions of faces of convex polytopes. The rank (or dimension) of an abstract polytope is determined by the length of the maximal ordered chains of its faces, and an abstract polytope of rank n is called an abstract n-polytope. For abstract polytopes of rank 2, this means that the elements of the partially ordered set are sets of vertices with either zero vertices (the empty set), one vertex, two vertices (an edge), or the entire vertex set, ordered by inclusion of sets, that each vertex belongs to exactly two edges, and that the undirected graph formed by the vertices and edges is connected. An abstract polytope is called an abstract apeirotope if it has infinitely many elements, and an abstract 2-apeirotope is called an abstract apeirogon.

Unschooling may emphasize free, undirected play as a major component of children's education. A fundamental premise of unschooling is that learning is a natural process constantly taking place and that curiosity is innate and children want to learn. From this an argument can be made that institutionalizing children in a so-called "one size fits all" or "factory model" school is an inefficient use of the children's time, because it requires each child to learn specific subject matter in a particular manner, at a particular pace, and at a specific time regardless of that individual's present or future needs, interests, goals, or any pre-existing knowledge they might have about the topic. Many unschoolers believe that opportunities for valuable hands-on, community-based, spontaneous, and real- world experiences may be missed when educational opportunities are limited to, or dominated by, those inside a school building.

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.

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.

She contributed two songs to the Valentine's Day soundtrack, including the country-pop song "Today Was a Fairytale", which became her first number one on the Canadian Hot 100 chart and her second number-two peaking song in the U.S. While filming her cinematic debut Valentine's Day in October 2009, Swift began a romantic relationship with co- star Taylor Lautner; they broke up later that year. The romantic comedy, released in 2010, saw her play the ditzy girlfriend of a high school jock, a role which the Los Angeles Times felt showed Swift had "serious comedic potential". On the other hand, in a scathing review, a critic for Variety deemed her "entirely undirected", arguing "she needs to find a skilled director to tamp her down and channel her obviously abundant energy". Swift made her TV acting debut in a 2009 episode of CBS's CSI: Crime Scene Investigation, playing a rebellious teenager.

Havens model a certain class of strategies for an evader in a pursuit-evasion game in which fewer than k pursuers attempt to capture a single evader, the pursuers and evader are both restricted to the vertices of a given undirected graph, and the positions of the pursuers and evader are known to both players. At each move of the game, a new pursuer may be added to an arbitrary vertex of the graph (as long as fewer than k pursuers are placed on the graph at any time) or one of the already- added pursuers may be removed from the graph. However, before a new pursuer is added, the evader is first informed of its new location and may move along the edges of the graph to any unoccupied vertex. While moving, the evader may not pass through any vertex that is already occupied by any of the pursuers.

In mathematics, a random minimum spanning tree may be formed by assigning random weights from some distribution to the edges of an undirected graph, and then constructing the minimum spanning tree of the graph. When the given graph is a complete graph on vertices, and the edge weights have a continuous distribution function whose derivative at zero is , then the expected weight of its random minimum spanning trees is bounded by a constant, rather than growing as a function of . More precisely, this constant tends in the limit (as goes to infinity) to , where is the Riemann zeta function and is Apéry's constant. For instance, for edge weights that are uniformly distributed on the unit interval, the derivative is , and the limit is just .. In contrast to uniformly random spanning trees of complete graphs, for which the typical diameter is proportional to the square root of the number of vertices, random minimum spanning trees of complete graphs have typical diameter proportional to the cube root.

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.

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.

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.

The Rado graph may also be formed by a construction resembling that for Paley graphs, taking as the vertices of a graph all the prime numbers that are congruent to 1 modulo 4, and connecting two vertices by an edge whenever one of the two numbers is a quadratic residue modulo the other. By quadratic reciprocity and the restriction of the vertices to primes congruent to 1 mod 4, this is a symmetric relation, so it defines an undirected graph, which turns out to be isomorphic to the Rado graph. Another construction of the Rado graph shows that it is an infinite circulant graph, with the integers as its vertices and with an edge between each two integers whose distance (the absolute value of their difference) belongs to a particular set S. To construct the Rado graph in this way, S may be chosen randomly, or by choosing the indicator function of S to be the concatenation of all finite binary sequences., Section 1.2.

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.

An undirected graph may be viewed as a simplicial complex with its vertices as zero-dimensional simplices and the edges as one-dimensional simplices.. The chain complex of this topological space consists of its edge space and vertex space (the Boolean algebra of sets of vertices), connected by a boundary operator that maps any spanning subgraph (an element of the edge space) to its set of odd-degree vertices (an element of the vertex space). The homology group :H_1(G,\Z_2) consists of the elements of the edge space that map to the zero element of the vertex space; these are exactly the Eulerian subgraphs. Its group operation is the symmetric difference operation on Eulerian subgraphs. Replacing \Z_2 in this construction by an arbitrary ring allows the definition of cycle spaces to be extended to cycle spaces with coefficients in the given ring, that form modules over the ring.. In particular, the integral cycle space is the space :H_1(G,\Z).

Conversely, every median graph G may be represented in this way as the solution set to a 2-satisfiability instance. To find such a representation, create a 2-satisfiability instance in which each variable describes the orientation of one of the edges in the graph (an assignment of a direction to the edge causing the graph to become directed rather than undirected) and each constraint allows two edges to share a pair of orientations only when there exists a vertex v such that both orientations lie along shortest paths from other vertices to v. Each vertex v of G corresponds to a solution to this 2-satisfiability instance in which all edges are directed towards v. Each solution to the instance must come from some vertex v in this way, where v is the common intersection of the sets Wuw for edges directed from w to u; this common intersection exists due to the Helly property of the sets Wuw.

Dejter showedDejter I. J. "Perfect domination in regular grid graphs", Austral. Jour. Combin., 42 (2008), 99-114 that there is an uncountable number of parallel total perfect codes in the planar integer lattice graph L; in contrast, there is just one 1-perfect code, and just one total perfect code in L, the latter code restricting to total perfect codes of rectangular grid graphs (which yields an asymmetric, Penrose, tiling of the plane); in particular, Dejter characterized all cycle products Cm x Cn containing parallel total perfect codes, and the d-perfect and total perfect code partitions of L and Cm x Cn, the former having as quotient graph the undirected Cayley graphs of the cyclic group of order 2d2+2d+1 with generator set {1,2d2}. In 2012, Araujo and DejterDejter I. J.; Araujo C. "Lattice-like total perfect codes", Discussiones Mathematicae Graph Theory, 34 (2014) 57–74, doi:10.7151/dmgt.1715.

One can define a binary relation on the edges of an arbitrary undirected graph, according to which two edges e and f are related if and only if either e = f or the graph contains a simple cycle through both e and f. Every edge is related to itself, and an edge e is related to another edge f if and only if f is related in the same way to e. Less obviously, this is a transitive relation: if there exists a simple cycle containing edges e and f, and another simple cycle containing edges f and g, then one can combine these two cycles to find a simple cycle through e and g. Therefore, this is an equivalence relation, and it can be used to partition the edges into equivalence classes, subsets of edges with the property that two edges are related to each other if and only if they belong to the same equivalence class.

For a finite set of labeled items, every pair of weak orderings may be connected to each other by a sequence of moves that add or remove one dichotomy at a time to or from this set of dichotomies. Moreover, the undirected graph that has the weak orderings as its vertices, and these moves as its edges, forms a partial cube.. Geometrically, the total orderings of a given finite set may be represented as the vertices of a permutohedron, and the dichotomies on this same set as the facets of the permutohedron. In this geometric representation, the weak orderings on the set correspond to the faces of all different dimensions of the permutohedron (including the permutohedron itself, but not the empty set, as a face). The codimension of a face gives the number of equivalence classes in the corresponding weak ordering.. In this geometric representation the partial cube of moves on weak orderings is the graph describing the covering relation of the face lattice of the permutohedron.

Animated example of a depth-first search For the following graph: alt=An undirected graph with edges AB, BD, BF, FE, AC, CG, AE a depth-first search starting at A, assuming that the left edges in the shown graph are chosen before right edges, and assuming the search remembers previously visited nodes and will not repeat them (since this is a small graph), will visit the nodes in the following order: A, B, D, F, E, C, G. The edges traversed in this search form a Trémaux tree, a structure with important applications in graph theory. Performing the same search without remembering previously visited nodes results in visiting nodes in the order A, B, D, F, E, A, B, D, F, E, etc. forever, caught in the A, B, D, F, E cycle and never reaching C or G. Iterative deepening is one technique to avoid this infinite loop and would reach all nodes.

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.

A graph with four connected subgraphs that, when contracted, form a complete graph. It has no five-vertex complete minor by Wagner's theorem, so its Hadwiger number is exactly four. In graph theory, the Hadwiger number of an undirected graph G is the size of the largest complete graph that can be obtained by contracting edges of G. Equivalently, the Hadwiger number h(G) is the largest number k for which the complete graph Kk is a minor of G, a smaller graph obtained from G by edge contractions and vertex and edge deletions. The Hadwiger number is also known as the contraction clique number of G or the homomorphism degree of G. It is named after Hugo Hadwiger, who introduced it in 1943 in conjunction with the Hadwiger conjecture, which states that the Hadwiger number is always at least as large as the chromatic number of G. The graphs that have Hadwiger number at most four have been characterized by .

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).

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.

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.

The prologue is a fictionalized account of The Shot Heard 'Round the World, a home run by Bobby Thomson on October 3, 1951, that won the National League pennant for the New York Giants against their cross-town rivals, the Brooklyn Dodgers. In DeLillo's account, the game-winning ball is caught by a young black fan named Cotter Martin, while J. Edgar Hoover, watching in the stands, is informed in the middle of the game of the first Soviet test of the hydrogen bomb. The remainder of the novel, comprising six parts and an epilogue, is a reverse chronological account of the life of Nick Shay, the man who ultimately ends up with the baseball, from his undirected existence as an executive of a waste management company in Arizona in the 1990s back to his childhood in the Bronx in the 1950s, though the non-linear narrative includes a large number of digressions and ancillary subplots. Part 1 takes place in 1992.

The clique complex can also be viewed as a topological space in which each clique of k vertices is represented by a simplex of dimension k − 1\. The 1-skeleton of X(G) (also known as the underlying graph of the complex) is an undirected graph with a vertex for every 1-element set in the family and an edge for every 2-element set in the family; it is isomorphic to G.. Clique complexes are also known as Whitney complexes, after Hassler Whitney. A Whitney triangulation or clean triangulation of a two-dimensional manifold is an embedding of a graph G onto the manifold in such a way that every face is a triangle and every triangle is a face. If a graph G has a Whitney triangulation, it must form a cell complex that is isomorphic to the Whitney complex of G. In this case, the complex (viewed as a topological space) is homeomorphic to the underlying manifold.

To create a skew-symmetric graph from an undirected graph G with a specified matching M, view G as a switch graph in which the edges at each vertex are partitioned into matched and unmatched edges; an alternating path in G is then a regular path in this switch graph and an alternating cycle in G is a regular cycle in the switch graph. generalized alternating path algorithms to show that the existence of a regular path between any two vertices of a skew-symmetric graph may be tested in linear time. Given additionally a non-negative length function on the edges of the graph that assigns the same length to any edge e and to σ(e), the shortest regular path connecting a given pair of nodes in a skew-symmetric graph with m edges and n vertices may be tested in time O(m log n). If the length function is allowed to have negative lengths, the existence of a negative regular cycle may be tested in polynomial time.

David Ellis investigated the behavior of researchers in the physical and social sciences, and engineers and research scientists through semi-structured interviews using a grounded theory approach, with a focus on describing the activities associated with information seeking rather than describing a process. Ellis' initial investigations produced six key activities within the information seeking process: # Starting (activities that form the information search) # Chaining (following references) # Browsing (semi-directed search) # Differentiating (filtering and selecting sources based on judgement of quality and relevance) # Monitoring (keeping track of developments in an area) # Extracting (systematic extraction of material of interest from sources) Later studies by Ellis (focusing on academic researchers in other disciplines) resulted in the addition of two more activities: #Verifying (checking accuracy) # Ending (a final search, checking all material covered) Choo, Detlor and Turnbull elaborated on Ellis' model by applying it to information searching on the web. Choo identified the key activities associated with Ellis in online searching episodes and connected them with four types of searching (undirected viewing, conditioned viewing, information search, and formal search).

A bucket queue can be used to maintain the vertices of an undirected graph, prioritized by their degrees, and repeatedly find and remove the vertex of minimum degree. This greedy algorithm can be used to calculate the degeneracy of a given graph. It takes linear time, with or without the optimization that maintains a lower bound on the minimum priority, because each vertex is found in time proportional to its degree and the sum of all vertex degrees is linear in the number of edges of the graph.. In Dijkstra's algorithm for shortest paths in positively-weighted directed graphs, a bucket queue can be used to obtain a time bound of , where is the number of vertices, is the number of edges, is the diameter of the network, and is the maximum (integer) link cost.. This variant of Dijkstra's algorithm is also known as Dial's algorithm, after Robert B. Dial, who published it in 1969. In this algorithm, the priorities will only span a range of width , so the modular optimization can be used to reduce the space to .

Hintikka's proposal was met with skepticism by a number of logicians because some first-order sentences like the one below appear to capture well enough the natural language Hintikka sentence. : [\forall x_1 \, \exists y_1 \, \forall x_2 \, \exists y_2\, \varphi (x_1, x_2, y_1, y_2)] \wedge [\forall x_2 \, \exists y_2 \, \forall x_1 \, \exists y_1\, \varphi (x_1, x_2, y_1, y_2)] where : \varphi (x_1, x_2, y_1, y_2) denotes : (V(x_1) \wedge T(x_2)) \rightarrow (R(x_1,y_1) \wedge R(x_2,y_2) \wedge H(y_1, y_2) \wedge H(y_2, y_1)) Although much purely theoretical debate followed, it wasn't until 2009 that some empirical tests with students trained in logic found that they are more likely to assign models matching the "bidirectional" first-order sentence rather than branching-quantifier sentence to several natural-language constructs derived from the Hintikka sentence. For instance students were shown undirected bipartite graphs--with squares and circles as vertices--and asked to say whether sentences like "more than 3 circles and more than 3 squares are connected by lines" were correctly describing the diagrams.

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.

In computer science, the Bron–Kerbosch algorithm is an enumeration algorithm for finding maximal cliques in an undirected graph. That is, it lists all subsets of vertices with the two properties that each pair of vertices in one of the listed subsets is connected by an edge, and no listed subset can have any additional vertices added to it while preserving its complete connectivity. The Bron–Kerbosch algorithm was designed by Dutch scientists Coenraad Bron and Joep Kerbosch, who published its description in 1973. Although other algorithms for solving the clique problem have running times that are, in theory, better on inputs that have few maximal independent sets, the Bron–Kerbosch algorithm and subsequent improvements to it are frequently reported as being more efficient in practice than the alternatives.. It is well-known and widely used in application areas of graph algorithms such as computational chemistry.. A contemporaneous algorithm of , although presented in different terms, can be viewed as being the same as the Bron–Kerbosch algorithm, as it generates the same recursive search tree..

Each cycle in the minimum weight cycle basis has a set of edges that are dual to the edges of one of the cuts in the Gomory–Hu tree. When cycle weights may be tied, the minimum-weight cycle basis may not be unique, but in this case it is still true that the Gomory–Hu tree of the dual graph corresponds to one of the minimum weight cycle bases of the graph.. In directed planar graphs, simple directed cycles are dual to directed cuts (partitions of the vertices into two subsets such that all edges go in one direction, from one subset to the other). Strongly oriented planar graphs (graphs whose underlying undirected graph is connected, and in which every edge belongs to a cycle) are dual to directed acyclic graphs in which no edge belongs to a cycle. To put this another way, the strong orientations of a connected planar graph (assignments of directions to the edges of the graph that result in a strongly connected graph) are dual to acyclic orientations (assignments of directions that produce a directed acyclic graph)..

In this more general context, the convex hull of a set S is the intersection of the family members that contain S, and the Radon number of a space is the smallest r such that any r points have two subsets whose convex hulls intersect. Similarly, one can define the Helly number h and the Carathéodory number c by analogy to their definitions for convex sets in Euclidean spaces, and it can be shown that these numbers satisfy the inequalities h < r ≤ ch + 1.. In an arbitrary undirected graph, one may define a convex set to be a set of vertices that includes every induced path connecting a pair of vertices in the set. With this definition, every set of ω + 1 vertices in the graph can be partitioned into two subsets whose convex hulls intersect, and ω + 1 is the minimum number for which this is possible, where ω is the clique number of the given graph.. For related results involving shortest paths instead of induced paths see and .

The paths through the intersection taken by traffic to get from an incoming lane to an outgoing lane may be represented as the edges of an undirected graph. For instance, this graph might have an edge from an incoming to an outgoing lane of traffic that both belong to the same segment of road, representing a U-turn from that segment back to that segment, only if U-turns are allowed at the junction. For a given subset of these edges, the subset represents a collection of paths that can all be traversed without interference from each other if and only if the subset does not include any pair of edges that would cross if the two edges were placed in a single page of a book embedding. Thus, a book embedding of this graph describes a partition of the paths into non-interfering subsets, and the book thickness of this graph (with its fixed embedding on the spine) gives the minimum number of distinct phases needed for a signalling schedule that includes all possible traffic paths through the junction..

Removing any two vertices (yellow) cannot disconnect a three-dimensional polyhedron: one can choose a third vertex (green), and a nontrivial linear function whose zero set (blue) passes through these three vertices, allowing connections from the chosen vertex to the minimum and maximum of the function, and from any other vertex to the minimum or maximum. In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional polyhedra and higher- dimensional polytopes. It states that, if one forms an undirected graph from the vertices and edges of a convex d-dimensional polyhedron or polytope (its skeleton), then the resulting graph is at least d-vertex-connected: the removal of any d − 1 vertices leaves a connected subgraph. For instance, for a three-dimensional polyhedron, even if two of its vertices (together with their incident edges) are removed, for any pair of vertices there will still exist a path of vertices and edges connecting the pair.. Balinski's theorem is named after mathematician Michel Balinski, who published its proof in 1961,.

As an example, the connectivity of an undirected graph can be expressed in MSO1 as the statement that, for every partition of the vertices into two nonempty subsets, there exists an edge from one subset to the other. A partition of the vertices can be described by the subset S of vertices on one side of the partition, and each such subset should either describe a trivial partition (one in which one or the other side is empty) or be crossed by an edge. That is, a graph is connected when it models the MSO1 formula :\forall S\Bigl( \forall x(x\in S) \vee \forall y\bigl(\lnot(y\in S)\bigr) \vee \exists x\exists y\bigl(x\in S\wedge \lnot(y\in S) \wedge x\sim y\bigr) \Bigr). However, connectivity cannot be expressed in first-order graph logic, nor can it be expressed in existential MSO1 (the fragment of MSO1 in which all set quantifiers are existential and occur at the beginning of the sentence) nor even existential MSO2.

A compatibility graph of partial words Two partial words are said to be compatible when they have the same length and when every position that is a non-wildcard in both of them has the same character in both. If one forms an undirected graph with a vertex for each partial word in a collection of partial words, and an edge for each compatible pair, then the cliques of this graph come from sets of partial words that all match at least one common string. This graph-theoretical interpretation of compatibility of partial words plays a key role in the proof of hardness of approximation of the clique problem, in which a collection of partial words representing successful runs of a probabilistically checkable proof verifier has a large clique if and only if there exists a valid proof of an underlying NP-complete problem. The faces (subcubes) of an n-dimensional hypercube can be described by partial words of length n over a binary alphabet, whose symbols are the Cartesian coordinates of the hypercube vertices (e.g.

Given a connected, undirected graph G, a shortest-path tree rooted at vertex v is a spanning tree T of G, such that the path distance from root v to any other vertex u in T is the shortest path distance from v to u in G. In connected graphs where shortest paths are well-defined (i.e. where there are no negative-length cycles), we may construct a shortest-path tree using the following algorithm: # Compute dist(u), the shortest-path distance from root v to vertex u in G using Dijkstra's algorithm or Bellman–Ford algorithm. # For all non-root vertices u, we can assign to u a parent vertex pu such that pu is connected to u, and that dist(pu) + edge_dist(pu,u) = dist(u). In case multiple choices for pu exist, choose pu for which there exists a shortest path from v to pu with as few edges as possible; this tie-breaking rule is needed to prevent loops when there exist zero-length cycles.

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.

A common way to form covering graphs uses voltage graphs, in which the darts of the given graph G (that is, pairs of directed edges corresponding to the undirected edges of G) are labeled with inverse pairs of elements from some group. The derived graph of the voltage graph has as its vertices the pairs (v,x) where v is a vertex of G and x is a group element; a dart from v to w labeled with the group element y in G corresponds to an edge from (v,x) to (w,xy) in the derived graph. The universal cover can be seen in this way as a derived graph of a voltage graph in which the edges of a spanning tree of the graph are labeled by the identity element of the group, and each remaining pair of darts is labeled by a distinct generating element of a free group. The bipartite double can be seen in this way as a derived graph of a voltage graph in which each dart is labeled by the nonzero element of the group of order two.

A graph that has the complete graph K4 as a 1-shallow minor. Each of the four vertex subsets indicated by the dashed rectangles induces a connected subgraph with radius one, and there exists an edge between every pair of subsets. One way of defining a minor of an undirected graph G is by specifying a subgraph H of G, and a collection of disjoint subsets Si of the vertices of G, each of which forms a connected induced subgraph Hi of H. The minor has a vertex vi for each subset Si, and an edge vivj whenever there exists an edge from Si to Sj that belongs to H. In this formulation, a d-shallow minor (alternatively called a shallow minor of depth d) is a minor that can be defined in such a way that each of the subgraphs Hi has radius at most d, meaning that it contains a central vertex ci that is within distance d of all the other vertices of Hi. Note that this distance is measured by hop count in Hi, and because of that it may be larger than the distance in G., Section 4.2 "Shallow Minors", pp. 62–65.

A gene co-expression network constructed from a microarray dataset containing gene expression profiles of 7221 genes for 18 gastric cancer patients A gene co-expression network (GCN) is an undirected graph, where each node corresponds to a gene, and a pair of nodes is connected with an edge if there is a significant co-expression relationship between them. Having gene expression profiles of a number of genes for several samples or experimental conditions, a gene co-expression network can be constructed by looking for pairs of genes which show a similar expression pattern across samples, since the transcript levels of two co-expressed genes rise and fall together across samples. Gene co-expression networks are of biological interest since co- expressed genes are controlled by the same transcriptional regulatory program, functionally related, or members of the same pathway or protein complex. The direction and type of co-expression relationships are not determined in gene co-expression networks; whereas in a gene regulatory network (GRN) a directed edge connects two genes, representing a biochemical process such as a reaction, transformation, interaction, activation or inhibition.

A graph with 6 vertices and 7 edges where the vertex number 6 on the far-left is a leaf vertex or a pendant vertex In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices.

They weight the edges of a grid graph by a numeric estimate of how visually apparent a seam across that edge would be, and find a bottleneck shortest path for these weights. Using this path as the seam, rather than a more conventional shortest path, causes their system to find a seam that is difficult to discern at all of its points, rather than allowing it to trade off greater visibility in one part of the image for lesser visibility elsewhere. A solution to the minimax path problem between the two opposite corners of a grid graph can be used to find the weak Fréchet distance between two polygonal chains. Here, each grid graph vertex represents a pair of line segments, one from each chain, and the weight of an edge represents the Fréchet distance needed to pass from one pair of segments to another.. If all edge weights of an undirected graph are positive, then the minimax distances between pairs of points (the maximum edge weights of minimax paths) form an ultrametric; conversely every finite ultrametric space comes from minimax distances in this way.

Because each entry in the adjacency matrix requires only one bit, it can be represented in a very compact way, occupying only 2/8 bytes to represent a directed graph, or (by using a packed triangular format and only storing the lower triangular part of the matrix) approximately 2/16 bytes to represent an undirected graph. Although slightly more succinct representations are possible, this method gets close to the information- theoretic lower bound for the minimum number of bits needed to represent all -vertex graphs.. For storing graphs in text files, fewer bits per byte can be used to ensure that all bytes are text characters, for instance by using a Base64 representation.. Besides avoiding wasted space, this compactness encourages locality of reference. However, for a large sparse graph, adjacency lists require less storage space, because they do not waste any space to represent edges that are not present. An alternative form of adjacency matrix (which, however, requires a larger amount of space) replaces the numbers in each element of the matrix with pointers to edge objects (when edges are present) or null pointers (when there is no edge).

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).

Optimal (span-5) radio coloring of a 6-cycle In graph theory, a branch of mathematics, a radio coloring of an undirected graph is a form of graph coloring in which one assigns positive integer labels to the graphs such that the labels of adjacent vertices differ by at least two, and the labels of vertices at distance two from each other differ by at least one. Radio coloring was first studied by , under a different name, -labeling.. It was called radio coloring by Frank Harary because it models the problem of channel assignment in radio broadcasting, while avoiding electromagnetic interference between radio stations that are near each other both in the graph and in their assigned channel frequencies. The span of a radio coloring is its maximum label, and the radio coloring number of a graph is the smallest possible span of a radio coloring.. See in particular Section 3, "Radio coloring". For instance, the graph consisting of two vertices with a single edge has radio coloring number 3: it has a radio coloring with one vertex labeled 1 and the other labeled 3, but it is not possible for a radio coloring of this graph to use only the labels 1 and 2.