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Simplicial Vertex
In graph theory, a simplicial vertex v is a vertex whose closed neighborhood N_ /math> in a graph G forms a clique A clique (AusE, CanE, or ; ), in the social sciences, is a small group of individuals who interact with one another and share similar interests rather than include others. Interacting with cliques is part of normative social development regardles ..., where every pair of neighbors is adjacent to each other. A vertex of a graph is bisimplicial if the set of it and its neighbours is the union of two cliques, and is -simplicial if the set is the union of cliques. A vertex is co-simplicial if its non-neighbours form an independent set. Addario-Berry et al. demonstrated that every even-hole-free graph (or more specifically, even-cycle-free graph, as 4-cycles are also excluded here) contains a bisimplicial vertex, which settled a conjecture by Reed. The proof was later shown to be flawed by Chudnovsky & Seymour, who gave a correct proof. Due to this property, the fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bisimplicial Vertex
In graph theory, a simplicial vertex v is a vertex (graph theory), vertex whose neighborhood (graph theory), closed neighborhood N_[v] in a graph G forms a clique (graph theory), clique, where every pair of neighbors is adjacent to each other. A vertex of a graph is bisimplicial if the set of it and its neighbours is the union of two cliques, and is -simplicial if the set is the union of cliques. A vertex is co-simplicial if its non-neighbours form an independent set (graph theory), independent set. Addario-Berry et al. demonstrated that every even-hole-free graph (or more specifically, even-cycle-free graph, as 4-cycles are also excluded here) contains a bisimplicial vertex, which settled a conjecture by Reed. The proof was later shown to be flawed by Chudnovsky & Seymour, who gave a correct proof. Due to this property, the family of all even-cycle-free graphs is χ-bounded, \chi-bounded. See also *Even-hole-free graph *χ-bounded, \chi-bounded family of graphs References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (graph Theory)
In discrete 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. A vertex ''w'' is said to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborhood (graph Theory)
In graph theory, an adjacent vertex of a vertex (graph theory), vertex in a Graph (discrete mathematics), graph is a vertex that is connected to by an edge (graph theory), edge. The neighbourhood of a vertex in a graph is the subgraph of induced subgraph, induced by all vertices adjacent to , i.e., the graph composed of the vertices adjacent to and all edges connecting vertices adjacent to . The neighbourhood is often denoted or (when the graph is unambiguous) . The same neighbourhood notation may also be used to refer to sets of adjacent vertices rather than the corresponding induced subgraphs. The neighbourhood described above does not include itself, and is more specifically the open neighbourhood of ; it is also possible to define a neighbourhood in which itself is included, called the closed neighbourhood and denoted by . When stated without any qualification, a neighbourhood is assumed to be open. Neighbourhoods may be used to represent graphs in computer algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique (graph Theory)
In graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinformatics. Definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Set (graph Theory)
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in S. A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening. A maximal independent set is an independent set that is not a proper subset of any other independent set. A maximum independent set is an independent set of largest possible size for a given graph G. This size is called the independence number of ''G'' and is usually denoted by \alpha(G). The optimization problem of finding such a set is called the maximum independent set problem. It is a strongly NP-hard problem. As ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. In 2020, most of the editorial board of ''JCTA'' resigned to form a new, |
Even-hole-free Graph
In the mathematical area of graph theory, a graph is even-hole-free if it contains no induced cycle with an even number of vertices. More precisely, the definition may allow the graph to have induced cycles of length four, or may also disallow them: the latter is referred to as even-cycle-free graphs. demonstrated that every even-cycle-free graph contains a bisimplicial vertex (a vertex whose neighborhood is the union of two cliques), which settled a conjecture by Reed. The proof was later shown to be flawed by , who gave a correct proof. Recognition gave the first polynomial time recognition algorithm for even-hole-free graphs, which runs in (n^) time. later improved this to (n^). and improved this to (n^) time. The best currently known algorithm is given by which runs in (n^9) time. While even-hole-free graphs can be recognized in polynomial time, it is NP-complete to determine whether a graph contains an even hole that includes a specific vertex. It is unknown whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
χ-bounded
In graph theory, a \chi-bounded family \mathcal of graphs is one for which there is some function f such that, for every integer t the graphs in \mathcal with t=\omega(G) ( clique number) can be colored with at most f(t) colors. The function f(t) is called a \chi-binding function for \mathcal. These concepts and their notations were formulated by András Gyárfás. The use of the Greek letter chi in the term \chi-bounded is based on the fact that the chromatic number of a graph G is commonly denoted \chi(G). An overview of the area can be found in a survey of Alex Scott and Paul Seymour. Nontriviality It is not true that the family of all graphs is \chi-bounded. As , and showed, there exist triangle-free graphs of arbitrarily large chromatic number, so for these graphs it is not possible to define a finite value of f(2). Thus, \chi-boundedness is a nontrivial concept, true for some graph families and false for others. Specific classes Every class of graphs of bounded chro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even-hole-free Graph
In the mathematical area of graph theory, a graph is even-hole-free if it contains no induced cycle with an even number of vertices. More precisely, the definition may allow the graph to have induced cycles of length four, or may also disallow them: the latter is referred to as even-cycle-free graphs. demonstrated that every even-cycle-free graph contains a bisimplicial vertex (a vertex whose neighborhood is the union of two cliques), which settled a conjecture by Reed. The proof was later shown to be flawed by , who gave a correct proof. Recognition gave the first polynomial time recognition algorithm for even-hole-free graphs, which runs in (n^) time. later improved this to (n^). and improved this to (n^) time. The best currently known algorithm is given by which runs in (n^9) time. While even-hole-free graphs can be recognized in polynomial time, it is NP-complete to determine whether a graph contains an even hole that includes a specific vertex. It is unknown whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
