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A maximum matching (also known as maximum-cardinality matching [2]) is a matching that contains the largest possible number of edges. There may be many maximum matchings. The matching number of a graph G is the size of a maximum matching. Every maximum matching is maximal, but not every maximal matching is a maximum 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.
In computer science and graph theory, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized. A special case of it is the assignment problem , in which the input is restricted to be a bipartite graph , and the matching constrained to be have cardinality that of the ...
Maximum cardinality matching, the problem solved by the algorithm, and its generalization to non-bipartite graphs; Assignment problem, a generalization of this problem on weighted graphs, solved e.g. by the Hungarian algorithm; Edmonds–Karp algorithm for finding maximum flow, a generalization of the Hopcroft–Karp algorithm
The matching problem can be generalized by assigning weights to edges in G and asking for a set M that produces a matching of maximum (minimum) total weight: this is the maximum weight matching problem. This problem can be solved by a combinatorial algorithm that uses the unweighted Edmonds's algorithm as a subroutine. [6]
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.
Therefore, by the Tutte–Berge formula, it has at most (1−3+16)/2 = 7 edges in any matching. In the mathematical discipline of graph theory the Tutte–Berge formula is a characterization of the size of a maximum matching in a graph.
Let M be a maximum matching and consider an alternating chain such that the edges in the path alternates between being and not being in M.If the alternating chain is a cycle or a path of even length starting on an unmatched vertex, then a new maximum matching M ′ can be found by interchanging the edges found in M and not in M.