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Gallai–Edmonds Decomposition
In graph theory, the Gallai–Edmonds decomposition is a partition of the vertices of a graph into three subsets which provides information on the structure of maximum matchings in the graph. Tibor Gallai and Jack Edmonds independently discovered it and proved its key properties. The Gallai–Edmonds decomposition of a graph can be found using the blossom algorithm. Properties Given a graph G, its Gallai–Edmonds decomposition consists of three disjoint sets of vertices, A(G), C(G), and D(G), whose union is V(G): the set of all vertices of G. First, the vertices of G are divided into ''essential vertices'' (vertices which are covered by every maximum matching in G) and ''inessential vertices'' (vertices which are left uncovered by at least one maximum matching in G). The set D(G) is defined to contain all the inessential vertices. Essential vertices are split into A(G) and C(G): the set A(G) is defined to contain all essential vertices adjacent to at least one vertex of D(G) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gallai–Edmonds Decomposition
In graph theory, the Gallai–Edmonds decomposition is a partition of the vertices of a graph into three subsets which provides information on the structure of maximum matchings in the graph. Tibor Gallai and Jack Edmonds independently discovered it and proved its key properties. The Gallai–Edmonds decomposition of a graph can be found using the blossom algorithm. Properties Given a graph G, its Gallai–Edmonds decomposition consists of three disjoint sets of vertices, A(G), C(G), and D(G), whose union is V(G): the set of all vertices of G. First, the vertices of G are divided into ''essential vertices'' (vertices which are covered by every maximum matching in G) and ''inessential vertices'' (vertices which are left uncovered by at least one maximum matching in G). The set D(G) is defined to contain all the inessential vertices. Essential vertices are split into A(G) and C(G): the set A(G) is defined to contain all essential vertices adjacent to at least one vertex of D(G) ... [...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] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' m ... [...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] |
Tibor Gallai
Tibor Gallai (born Tibor Grünwald, 15 July 1912 – 2 January 1992) was a Hungarian mathematician. He worked in combinatorics, especially in graph theory, and was a lifelong friend and collaborator of Paul Erdős. He was a student of Dénes Kőnig and an advisor of László Lovász. He was a corresponding member of the Hungarian Academy of Sciences (1991). His main results The Edmonds–Gallai decomposition theorem, which was proved independently by Gallai and Jack Edmonds, describes finite graphs from the point of view of matchings. Gallai also proved, with Milgram, Dilworth's theorem in 1947, but as they hesitated to publish the result, Dilworth independently discovered and published it.P. ErdősIn memory of Tibor Gallai ''Combinatorica'', 12(1992), 373–374. Gallai was the first to prove the higher-dimensional version of van der Waerden's theorem. With Paul Erdős he gave a necessary and sufficient condition for a sequence to be the degree sequence of a graph, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jack Edmonds
Jack R. Edmonds (born April 5, 1934) is an American-born and educated computer scientist and mathematician who lived and worked in Canada for much of his life. He has made fundamental contributions to the fields of combinatorial optimization, polyhedral combinatorics, discrete mathematics and the theory of computing. He was the recipient of the 1985 John von Neumann Theory Prize. Early career Edmonds attended Duke University before completing his undergraduate degree at George Washington University in 1957. He thereafter received a master's degree in 1960 at the University of Maryland under Bruce L. Reinhart with a thesis on the problem of embedding graphs into surfaces. From 1959 to 1969 he worked at the National Institute of Standards and Technology (then the National Bureau of Standards), and was a founding member of Alan J. Goldman, Alan Goldman’s newly created Operations Research Section in 1961. Goldman proved to be a crucial influence by enabling Edmonds to work in a RAND ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blossom Algorithm
In graph theory, the blossom algorithm is an algorithm for constructing maximum matchings on graphs. The algorithm was developed by Jack Edmonds in 1961, and published in 1965. Given a general graph , the algorithm finds a matching such that each vertex in is incident with at most one edge in and is maximized. The matching is constructed by iteratively improving an initial empty matching along augmenting paths in the graph. Unlike bipartite matching, the key new idea is that an odd-length cycle in the graph (blossom) is contracted to a single vertex, with the search continuing iteratively in the contracted graph. The algorithm runs in time , where is the number of edges of the graph and is its number of vertices. A better running time of O( , E, \sqrt ) for the same task can be achieved with the much more complex algorithm of Micali and Vazirani. A major reason that the blossom algorithm is important is that it gave the first proof that a maximum-size matching could be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Set
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. Generalizations This definition of disjoint sets can be extended to a family of sets \left(A_i\right)_: the family is pairwise disjoint, or mutually disjoint if A_i \cap A_j = \varnothing whenever i \neq j. Alternatively, some authors use the term disjoint to refer to this notion as well. For families the notion of pairwise disjoint or mutually disjoint is sometimes defined in a subtly different manner, in that repeated identical members are allowed: the family is pairwise disjoint if A_i \cap A_j = \varnothing whenever A_i \neq A_j (every two ''distinct'' sets in the family are disjoint).. For example, the collection of sets is di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union (set Theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of zero (0) sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols. Union of two sets The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In set-builder notation, :A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup B = \ As another example, the number 9 is ''not'' contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even. Sets cannot have duplicate elements, so the union of the sets and is . Multip ... [...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] |
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. Dynamic co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factor-critical Graph
In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph.) is a graph with vertices in which every subgraph of vertices has a perfect matching. (A perfect matching in a graph is a subset of its edges with the property that each of its vertices is the endpoint of exactly one of the edges in the subset.) A matching that covers all but one vertex of a graph is called a near-perfect matching. So equivalently, a factor-critical graph is a graph in which there are near-perfect matchings that avoid every possible vertex. Examples Any odd-length cycle graph is factor-critical, as is any complete graph with an odd number of vertices. More generally, every Hamiltonian graph with an odd number of vertices is factor-critical. The friendship graphs (graphs formed by connecting a collection of triangles at a single common vertex) provide examples of graphs that are factor-critical but not Hamiltonian. If a graph is factor-critical, then so is the Myciels ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
