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Line Perfect Graph
In graph theory, a line perfect graph is a graph whose line graph is a perfect graph. Equivalently, these are the graphs in which every odd-length simple cycle is a triangle. A graph is line perfect if and only if each of its biconnected components is a bipartite graph, the complete graph , or a triangular book . Because these three types of biconnected component are all perfect graphs themselves, every line perfect graph is itself perfect. By similar reasoning, every line perfect graph is a parity graph, a Meyniel graph, and a perfectly orderable graph. Line perfect graphs generalize the bipartite graphs, and share with them the properties that the maximum matching and minimum vertex cover have the same size, and that the chromatic index equals the maximum degree. See also *Strangulated graph, a graph in which every peripheral cycle is a triangle References {{reflist, refs= {{citation , last = de Werra , first = D. , doi = 10.1007/BF01609025 , issue = 2 , journa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Perfect Graph
In graph theory, a line perfect graph is a graph whose line graph is a perfect graph. Equivalently, these are the graphs in which every odd-length simple cycle is a triangle. A graph is line perfect if and only if each of its biconnected components is a bipartite graph, the complete graph , or a triangular book . Because these three types of biconnected component are all perfect graphs themselves, every line perfect graph is itself perfect. By similar reasoning, every line perfect graph is a parity graph, a Meyniel graph, and a perfectly orderable graph. Line perfect graphs generalize the bipartite graphs, and share with them the properties that the maximum matching and minimum vertex cover have the same size, and that the chromatic index equals the maximum degree. See also *Strangulated graph, a graph in which every peripheral cycle is a triangle References {{reflist, refs= {{citation , last = de Werra , first = D. , doi = 10.1007/BF01609025 , issue = 2 , journa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meyniel Graph
In graph theory, a Meyniel graph is a graph in which every odd cycle of length five or more has at least two chords (edges connecting non-consecutive vertices of the cycle). The chords may be uncrossed (as shown in the figure) or they may cross each other, as long as there are at least two of them. The Meyniel graphs are named after Henri Meyniel (also known for Meyniel's conjecture), who proved that they are perfect graphs in 1976,. long before the proof of the strong perfect graph theorem completely characterized the perfect graphs. The same result was independently discovered by .. Perfection The Meyniel graphs are a subclass of the perfect graphs. Every induced subgraph of a Meyniel graph is another Meyniel graph, and in every Meyniel graph the size of a maximum clique equals the minimum number of colors needed in a graph coloring. Thus, the Meyniel graphs meet the definition of being a perfect graph, that the clique number equals the chromatic number in every induced su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Programming
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peripheral Cycle
In graph theory, a peripheral cycle (or peripheral circuit) in an undirected graph is, intuitively, a cycle that does not separate any part of the graph from any other part. Peripheral cycles (or, as they were initially called, peripheral polygons, because Tutte called cycles "polygons") were first studied by , and play important roles in the characterization of planar graphs and in generating the cycle spaces of nonplanar graphs. Definitions A peripheral cycle C in a graph G can be defined formally in one of several equivalent ways: *C is peripheral if it is a simple cycle in a connected graph with the property that, for every two edges e_1 and e_2 in G\setminus C, there exists a path in G that starts with e_1, ends with e_2, and has no interior vertices belonging to C.. *C is peripheral if it is an induced cycle with the property that the subgraph G\setminus C formed by deleting the edges and vertices of C is connected. *If C is any subgraph of G, a ''bridge'' of C is a mini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strangulated Graph
In graph theoretic mathematics, a strangulated graph is a graph in which deleting the edges of any induced cycle of length greater than three would disconnect the remaining graph. That is, they are the graphs in which every peripheral cycle is a triangle. Examples In a maximal planar graph, or more generally in every polyhedral graph, the peripheral cycles are exactly the faces of a planar embedding of the graph, so a polyhedral graph is strangulated if and only if all the faces are triangles, or equivalently it is maximal planar. Every chordal graph is strangulated, because the only induced cycles in chordal graphs are triangles, so there are no longer cycles to delete. Characterization A clique-sum of two graphs is formed by identifying together two equal-sized cliques in each graph, and then possibly deleting some of the clique edges. For the version of clique-sums relevant to strangulated graphs, the edge deletion step is omitted. A clique-sum of this type between two str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Degree
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B C D E F G H I K L M N O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Index
In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most different colors, for a given value of , or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree or . For some graphs, such as bipartite graphs and high-degree planar graphs, the number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Vertex Cover
In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. In computer science, the problem of finding a minimum vertex cover is a classical optimization problem. It is NP-hard, so it cannot be solved by a polynomial-time algorithm if P ≠ NP. Moreover, it is hard to approximate – it cannot be approximated up to a factor smaller than 2 if the unique games conjecture is true. On the other hand, it has several simple 2-factor approximations. It is a typical example of an NP-hard optimization problem that has an approximation algorithm. Its decision version, the vertex cover problem, was one of Karp's 21 NP-complete problems and is therefore a classical NP-complete problem in computational complexity theory. Furthermore, the vertex cover problem is fixed-parameter tractable and a central problem in parameterized complexity theory. The minimum vertex cover problem can be formulated as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Matching
Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph , and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible. An important special case of the maximum cardinality matching problem is when is a bipartite graph, whose vertices are partitioned between left vertices in and right vertices in , and edges in always connect a left vertex to a right vertex. In this case, the problem can be efficiently solved with simpler algorithms than in the general case. Algorithms for bipartite graphs Flow-based algorithm The simplest way to compute a maximum cardinality matching is to follow the Ford–Fulkerson algorithm. This algorithm solves the more general problem o ... [...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] |
Parity Graph
In graph theory, a parity graph is a graph in which every two induced paths between the same two vertices have the same parity: either both paths have odd length, or both have even length.Parity graphs Information System on Graph Classes and their Inclusions, retrieved 2016-09-25. This class of graphs was named and first studied by .. Related classes of graphs Parity graphs include the s, in which every two induced paths between the same two vertices have the same length. They also include the bipartite graphs, which may be charact ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''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 of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
