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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] |
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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] |
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Forbidden Induced Subgraph
In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor. A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph and the complete bipartite graph . For Kuratowski's theorem, the notion of containment is that of graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of at least one of these two graphs as a subgraph (in which case it does not belong to the planar graphs). Definition More generally, a forbidden g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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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] |
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Rooted Tree
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,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. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Trivially Perfect Graph
In graph theory, a trivially perfect graph is a graph with the property that in each of its induced subgraphs the size of the maximum independent set equals the number of maximal cliques. Trivially perfect graphs were first studied by but were named by ; Golumbic writes that "the name was chosen since it is trivial to show that such a graph is perfect graph, perfect." Trivially perfect graphs are also known as comparability graphs of trees, arborescent comparability graphs, and quasi-threshold graphs. Equivalent characterizations Trivially perfect graphs have several other equivalent characterizations: *They are the comparability graphs of Tree (set theory), order-theoretic trees. That is, let be a partial order such that for each , the set is well-ordered by the Relation (mathematics), relation , and also possesses a minimum element . Then the comparability graph of is trivially perfect, and every trivially perfect graph can be formed in this way. *They are the graphs that do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |