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Split Graph
In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by , and independently introduced by . A split graph may have more than one partition into a clique and an independent set; for instance, the path is a split graph, the vertices of which can be partitioned in three different ways: #the clique and the independent set #the clique and the independent set #the clique and the independent set Split graphs can be characterized in terms of their forbidden induced subgraphs: a graph is split if and only if no induced subgraph is a cycle on four or five vertices, or a pair of disjoint edges (the complement of a 4-cycle). Relation to other graph families From the definition, split graphs are clearly closed under complementation. Another characterization of split graphs involves complementation: they are chordal graphs the complements of which are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Split Graph
In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by , and independently introduced by . A split graph may have more than one partition into a clique and an independent set; for instance, the path is a split graph, the vertices of which can be partitioned in three different ways: #the clique and the independent set #the clique and the independent set #the clique and the independent set Split graphs can be characterized in terms of their forbidden induced subgraphs: a graph is split if and only if no induced subgraph is a cycle on four or five vertices, or a pair of disjoint edges (the complement of a 4-cycle). Relation to other graph families From the definition, split graphs are clearly closed under complementation. Another characterization of split graphs involves complementation: they are chordal graphs the complements of which are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Graph
In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph (clique number). Equivalently stated in symbolic terms an arbitrary graph G=(V,E) is perfect if and only if for all S\subseteq V we have \chi(G =\omega(G . The perfect graphs include many important families of graphs and serve to unify results relating colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time. In addition, several important min-max theorems in combinatorics, such as Dilworth's theorem, can be expressed in terms of the perfection of certain associated graphs. A graph G is 1-perfect if and only if \chi(G)=\omega(G). Then, G is perfect if and only if every induced subgraph of G is 1-perfect. Properties * By the perfect graph theorem, a graph G is perfect if and on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Independent Set
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 such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborhood (graph Theory)
In graph theory, an adjacent vertex of a vertex in a graph is a vertex that is connected to by an edge. The neighbourhood of a vertex in a graph is the subgraph of 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 algorithms, via the adjacency list and adjacency matrix representations. Neighbourhoods are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Clique
In the mathematical area of 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 bioinf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cocoloring
In graph theory, a cocoloring of a graph ''G'' is an assignment of colors to the vertices such that each color class forms an independent set in ''G'' or in the complement of ''G''. The cochromatic number z(''G'') of ''G'' is the fewest colors needed in any cocolorings of ''G''. The graphs with cochromatic number 2 are exactly the bipartite graphs, complements of bipartite graphs, and split graphs. As the requirement that each color class be a clique or independent is weaker than the requirement for coloring (in which each color class must be an independent set) and stronger than for subcoloring (in which each color class must be a disjoint union of cliques), it follows that the cochromatic number of ''G'' is less than or equal to the chromatic number In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of colorin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew-merged Permutation
In the theory of permutation patterns, a skew-merged permutation is a permutation that can be partitioned into an increasing sequence and a decreasing sequence. They were first studied by and given their name by . Characterization The two smallest permutations that cannot be partitioned into an increasing and a decreasing sequence are 3412 and 2143. was the first to establish that a skew-merged permutation can also be equivalently defined as a permutation that avoids the two patterns 3412 and 2143. A permutation is skew-merged if and only if its associated permutation graph is a split graph, a graph that can be partitioned into a clique (corresponding to the descending subsequence) and an independent set (corresponding to the ascending subsequence). The two forbidden patterns for skew-merged permutations, 3412 and 2143, correspond to two of the three forbidden induced subgraphs for split graphs, a four-vertex cycle and a graph with two disjoint edges, respectively. The third forb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Graph
In the mathematical field of graph theory, a permutation graph is a graph whose vertices represent the elements of a permutation, and whose edges represent pairs of elements that are reversed by the permutation. Permutation graphs may also be defined geometrically, as the intersection graphs of line segments whose endpoints lie on two parallel lines. Different permutations may give rise to the same permutation graph; a given graph has a unique representation (up to permutation symmetry) if it is prime with respect to the modular decomposition. Definition and characterization If \rho = (\sigma_1,\sigma_2,...,\sigma_n) is any permutation of the numbers from 1 to n, then one may define a permutation graph from \sigma in which there are n vertices v_1, v_2, ..., v_n, and in which there is an edge v_i v_j for any two indices i and j for which i\sigma_j. That is, two indices i and j determine an edge in the permutation graph exactly when they determine an inversion in the permutatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Threshold Graph
In graph theory, a threshold graph is a graph that can be constructed from a one-vertex graph by repeated applications of the following two operations: # Addition of a single isolated vertex to the graph. # Addition of a single dominating vertex to the graph, i.e. a single vertex that is connected to all other vertices. For example, the graph of the figure is a threshold graph. It can be constructed by beginning with a single-vertex graph (vertex 1), and then adding black vertices as isolated vertices and red vertices as dominating vertices, in the order in which they are numbered. Threshold graphs were first introduced by . A chapter on threshold graphs appears in , and the book is devoted to them. Alternative definitions An equivalent definition is the following: a graph is a threshold graph if there are a real number S and for each vertex v a real vertex weight w(v) such that for any two vertices v,u, uv is an edge if and only if w(u)+w(v)> S. Another equivalent definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cograph
In graph theory, a cograph, or complement-reducible graph, or ''P''4-free graph, is a graph that can be generated from the single-vertex graph ''K''1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes ''K''1 and is closed under complementation and disjoint union. Cographs have been discovered independently by several authors since the 1970s; early references include , , , and . They have also been called D*-graphs, hereditary Dacey graphs (after the related work of James C. Dacey Jr. on orthomodular lattices), and 2-parity graphs. They have a simple structural decomposition involving disjoint union and complement graph operations that can be represented concisely by a labeled tree, and used algorithmically to efficiently solve many problems such as finding the maximum clique that are hard on more general graph classes. Special cases of the cographs include the complete graphs, complete bipartite graphs, cluster gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
