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In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If the set of vertices is V = { 1 , 2 , … , n } {\displaystyle V=\{1,2,\dots ,n\}} then the Tutte matrix is an n -by- n matrix A with entries
In graph theory, a branch of mathematics, the matching preclusion number of a graph G (denoted mp(G)) is the minimum number of edges whose deletion results in the elimination of all perfect matchings or near-perfect matchings (matchings that cover all but one vertex in a graph with an odd number of vertices). [1]
Let G = (V,E) be a graph, where V are the vertices and E are the edges. A matching in G is a subset M of E, such that each vertex in V is adjacent to at most a single edge in M. A maximum matching is a matching of maximum cardinality. An edge e in E is called maximally matchable (or allowed) if there exists a maximum matching M that contains e.
In this graph, removing one vertex in the center produces three odd components, the three five-vertex lobes of the graph. Therefore, by the Tutte–Berge formula, it has at most (1−3+16)/2 = 7 edges in any matching.
Maximum cardinality matching is a fundamental problem in graph theory. [1] We are given a graph G, 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 ...
For example, this 2-uniform hypergraph represents a graph with 4 vertices {1,2,3,4} and 3 edges: { {1,3}, {1,4}, {2,4} } By the above definition, a matching in a graph is a set M of edges, such that each two edges in M have an empty intersection.
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In graph theory, a priority matching (also called: maximum priority matching) is a matching that maximizes the number of high-priority vertices that participate in the matching. Formally, we are given a graph G = ( V , E ) , and a partition of the vertex-set V into some k subsets, V 1 , …, V k , called priority classes .