|
Greedy Coloring
In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not in general use the minimum number of colors possible. Different choices of the sequence of vertices will typically produce different colorings of the given graph, so much of the study of greedy colorings has concerned how to find a good ordering. There always exists an ordering that produces an optimal coloring, but although such orderings can be found for many special classes of graphs, they are hard to find in general. Commonly used strategies for vertex ordering involve placing higher-degree vertices earlier than lower-degree vertices, or choosing vertices with fewer available colors in preference to vertices that are less con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greedy Colourings
Greedy may refer to: __NOTOC__ Music * ''Greedy'' (album), a 1997 album by Headless Chickens * "Greedy" (song), a song on the album ''Dangerous Woman'' by Ariana Grande * "Greedy", a single by Canadian rock band Pure * "Greedy", a song on the album ''Mass Nerder'' by the punk rock band All * "Greedy", a song on the album ''Lyfe 268‒192'' by Lyfe Jennings * "Greedy", a song on the album ''Training Day'' by The Away Team * "Greedy", a song on the mixtape ''Keep Flexin'' by Rich the Kid People * Greedy Smith (1956–2019), pseudonym of Andrew McArthur Smith, singer, musician and songwriter with the Australian pop/new wave band Mental As Anything * Greedy Williams (born 1997), American football player nicknamed "Greedy" * Jack Greedy (1929–1988), Canadian racing driver, member of the Canadian Motorsport Hall of Fame Other uses * Greedy algorithm, any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage * ''Greedy'' (film), a 199 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Independent Set
In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no 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 with three vertices , , and , and two edges and , the sets and are both maximally 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. The graphs in which all maximal independent sets have the same siz ... [...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] |
Distance-hereditary Graph
In graph theory, a branch of discrete mathematics, a distance-hereditary graph (also called a completely separable graph) is a graph in which the distances in any connected induced subgraph are the same as they are in the original graph. Thus, any induced subgraph inherits the distances of the larger graph. Distance-hereditary graphs were named and first studied by , although an equivalent class of graphs was already shown to be perfect in 1970 by Olaru and Sachs. It has been known for some time that the distance-hereditary graphs constitute an intersection class of graphs, but no intersection model was known until one was given by . Definition and characterization The original definition of a distance-hereditary graph is a graph such that, if any two vertices and belong to a connected induced subgraph of , then some shortest path connecting and in must be a subgraph of , so that the distance between and in is the same as the distance in . Distance-hereditary graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparability Graph
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, 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, apply transitive closure, and remove orientation. Equivalently, a comparability graph is a graph that has a transitive orientation, an assignment of directions to the edges of the graph (i.e. an orientation of the graph) such that the adjacency relation of the resulting directe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfectly Orderable Graph
In graph theory, a perfectly orderable graph is a graph whose vertices can be ordered in such a way that a greedy coloring algorithm with that ordering optimally colors every induced subgraph of the given graph. Perfectly orderable graphs form a special case of the perfect graphs, and they include the chordal graphs, comparability graphs, and distance-hereditary graphs. However, testing whether a graph is perfectly orderable is NP-complete. Definition The greedy coloring algorithm, when applied to a given ordering of the vertices of a graph ''G'', considers the vertices of the graph in sequence and assigns each vertex its first available color, the minimum excluded value for the set of colors used by its neighbors. Different vertex orderings may lead this algorithm to use different numbers of colors. There is always an ordering that leads to an optimal coloring – this is true, for instance, of the ordering determined from an optimal coloring by sorting the vertices by their color ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Subgraph
In the mathematical field of 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\subset 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique (graph Theory)
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] |
Perfect Elimination Ordering
In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs. or triangulated graphs.. Chordal graphs are a subset of the perfect graphs. They may be recognized in linear time, and several problems that are hard on other classes of graphs such as graph coloring may be solved in polynomial time when the input is chordal. The treewidth of an arbitrary graph may be characterized by the size of the cliques in the chordal graphs that contain it. Perfect elimination and efficient recogni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chordal Graph
In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a ''chord'', which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs. or triangulated graphs.. Chordal graphs are a subset of the perfect graphs. They may be recognized in linear time, and several problems that are hard on other classes of graphs such as graph coloring may be solved in polynomial time when the input is chordal. The treewidth of an arbitrary graph may be characterized by the size of the cliques in the chordal graphs that contain it. Perfect elimination and efficient recognit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. They can be recognized in linear time, and an optimal graph coloring or maximum clique in these graphs can be found in linear time. The interval graphs include all proper interval graphs, graphs defined in the same way from a set of unit intervals. These graphs have been used to model food webs, and to study scheduling problems in which one must select a subset of tasks to be performed at non-overlapping times. Other applications include assembling contiguous subsequences in DNA mapping, and temporal reasoning. Definition An interval graph is an undirected graph formed from a family of intervals :S_i,\quad i=0,1,2,\dots by creating one vertex for each interval , and connecting two ver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hard
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard problem is the subset sum problem. A more precise specification is: a problem ''H'' is NP-hard when every problem ''L'' in NP can be reduced in polynomial time to ''H''; that is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve any NP-hard problem would give polynomial time algorithms for all the problems in NP. As it is suspected that P≠NP, it is unlikely that such an algorithm exists. It is suspected that there are no polynomial-time algorithms for NP-hard problems, but that has not been proven. Moreover, the class P, in which all problems can be solved in polynomial time, is contained in the NP class. Defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
