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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 a maximum clique that are hard on more general graph classes. Special types of cograph include complete graphs, complete bipartite graphs, cluster graphs, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Comparability Graph
In graph theory and order 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, and divisor graphs. An incomparability graph is an undirected graph that connects pairs of elements that are not comparable to each other in a partial order. Definitions and characterization For any strict partially ordered set , the comparability graph of is the graph of which the vertices are the elements of and the edges are those pairs of elements such that . That is, for a partially ordered set, take the directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one ..., apply t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Complement Graph
In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there.. The complement is not the set complement of the graph; only the edges are complemented. Definition Let be a simple graph and let consist of all 2-element subsets of . Then is the complement of , where is the relative complement of in . For directed graphs, the complement can be defined in the same way, as a directed graph on the same vertex set, using the set of all 2-element ordered pairs of in place of the set in the formula above. In terms of the adjacency matrix ''A'' of the graph, if ''Q'' is the adjacency matrix of the complete graph of the same number of vertices (i.e. all entries ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Maximal Independent Set
In graph theory, a maximal independent set (MIS) or maximal stable set is an Independent set (graph theory), independent set that is not a subset of any other independent set. In other words, there is no Vertex (graph theory), vertex outside the independent set that may join it because it is maximal with respect to the independent set property. For example, in the graph , a Path graph, path with three vertices , , and , and two edges and , the sets and are both maximal independent. The set is independent, but is not maximal independent, because it is a subset of the larger independent set In this same graph, the maximal cliques are the sets and A MIS is also a dominating set in the graph, and every dominating set that is independent must be maximal independent, so MISs are also called independent dominating sets. A graph may have many MISs of widely varying sizes; the largest, or possibly several equally large, MISs of a graph is called a maximum independent set, ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Complement Graph
In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there.. The complement is not the set complement of the graph; only the edges are complemented. Definition Let be a simple graph and let consist of all 2-element subsets of . Then is the complement of , where is the relative complement of in . For directed graphs, the complement can be defined in the same way, as a directed graph on the same vertex set, using the set of all 2-element ordered pairs of in place of the set in the formula above. In terms of the adjacency matrix ''A'' of the graph, if ''Q'' is the adjacency matrix of the complete graph of the same number of vertices (i.e. all entries ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Clique-width
In graph theory, the clique-width of a graph is a parameter that describes the structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be small for dense graphs. It is defined as the minimum number of labels needed to construct by means of the following 4 operations : #Creation of a new vertex with label (denoted by ) #Disjoint union of two labeled graphs and (denoted by G \oplus H) #Joining by an edge every vertex labeled to every vertex labeled (denoted by ), where #Renaming label to label (denoted by ) Graphs of bounded clique-width include the cographs and distance-hereditary graphs. Although it is NP-hard to compute the clique-width when it is unbounded, and unknown whether it can be computed in polynomial time when it is bounded, efficient approximation algorithms for clique-width are known. Based on these algorithms and on Courcelle's theorem, many graph optimization problems that are NP-hard for arbitrary graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Turan 13-4
Turan (; ; , , ) is a historical region in Central Asia. The term is of Iranian origin and may refer to a particular prehistoric human settlement, a historic geographical region, or a culture. The original Turanians were an Iranian tribe of the Avestan age. Overview In ancient Iranian mythology, Tūr or Turaj (''Tuzh'' in Middle Persian) is the son of the emperor Fereydun. According to the account in the ''Shahnameh'', the nomadic tribes who inhabited these lands were ruled by Tūr. In that sense, the Turanians could be members of two Iranian peoples both descending from Fereydun, but with different geographical domains and often at war with each other. Turan, therefore, comprised five areas: the Kopet Dag region, the Atrek valley, parts of Bactria, Sogdia and Margiana. A later association of the original Turanians with Turkic peoples is based primarily on the subsequent Turkification of Central Asia, including the above areas. According to C. E. Bosworth, however ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Series-parallel Partial Order
In order theory, order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two simple composition operations... The series-parallel partial orders may be characterized as the N-free finite partial orders; they have order dimension at most two.. They include weak orders and the reachability relationship in Tree (graph theory), directed trees and directed series–parallel graphs. The comparability graphs of series-parallel partial orders are cographs. Series-parallel partial orders have been applied in job shop scheduling, machine learning of event sequencing in time series data, transmission sequencing of multimedia data, and throughput maximization in dataflow programming. Series-parallel partial orders have also been called multitrees;. however, that name is ambiguous: multitrees also refer to partial orders with no four-element diamond suborder and to other structures formed from multiple tree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Diameter (graph Theory)
In graph theory, the diameter of a connected undirected graph is the farthest distance between any two of its vertices. That is, it is the diameter of a set for the set of vertices of the graph, and for the shortest-path distance in the graph. Diameter may be considered either for weighted or for unweighted graphs. Researchers have studied the problem of computing the diameter, both in arbitrary graphs and in special classes of graphs. The diameter of a disconnected graph may be defined to be infinite, or undefined. Graphs of low diameter The degree diameter problem seeks tight relations between the diameter, number of vertices, and degree of a graph. One way of formulating it is to ask for the largest graph with given bounds on its degree and diameter. For any fixed degree, this maximum size is exponential in diameter, with the base of the exponent depending on the degree. The girth of a graph, the length of its shortest cycle, can be at most 2k+1 for a graph of diameter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Connected Component (graph Theory)
In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are sometimes called connected components. The number of components in a given graph is an important graph invariant, and is closely related to invariants of matroids, topological spaces, and matrices. In random graphs, a frequently occurring phenomenon is the incidence of a giant component, one component that is significantly larger than the others; and of a percolation threshold, an edge probability above which a giant component exists and below which it does not. The components of a graph can be constructed in linear time, and a special case of the problem, connected-component labeling, is a basic technique in image analysis. Dy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Chordal Completion
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 . Chordal completions can also be used to characterize several other graph classes including AT-free graphs, claw-free AT-free graphs, and cographs. The minimum chordal completion was one of twelve computational problems whose complexity was listed as open in the 1979 book ''Computers and Intractability''. Applications of chordal completion include modeling the problem of minimizing fill-in when performing Gaussian elimination on sparse sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Separable Permutation
In combinatorics, combinatorial mathematics, a separable permutation is a permutation that can be obtained from the trivial permutation 1 by Direct sum of permutations, direct sums and Skew sum of permutations, skew sums. Separable permutations may be characterized by the forbidden permutation patterns 2413 and 3142;; , Theorem 2.2.36, p. p.58. they are also the permutations whose permutation graphs are cographs and the permutations that Order dimension, realize the series-parallel partial orders. It is possible to test in polynomial time whether a given separable permutation is a pattern in a larger permutation, or to find the longest common subpattern of two separable permutations. Definition and characterization define a separable permutation to be a permutation that has a ''separating tree'': a rooted binary tree in which the elements of the permutation appear (in permutation order) at the leaves of the tree, and in which the descendants of each tree node form a contiguou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Induced Subgraph
In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges, from the original graph, connecting pairs of vertices in that subset. Definition Formally, let G=(V,E) be any graph, and let S\subseteq V be any subset of vertices of . Then the induced subgraph G is the graph whose vertex set is S and whose edge set consists of all of the edges in E that have both endpoints in S . That is, for any two vertices u,v\in S , u and v are adjacent in G if and only if they are adjacent in G . The same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph G may also be called the subgraph induced in G by S , or (if context makes the choice of G unambiguous) the induced subgraph of S . Examples Important types of induced subgraphs include the following. * Induced paths are induced subgraphs that are paths. The shortest path between any two vertices in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
