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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.
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.
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]
An example of a bipartite graph, with a maximum matching (blue) and minimum vertex cover (red) both of size six. In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs.
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In this case, by the Tutte–Berge formula, the size of the matching is | | /; that is, the maximum matching is a perfect matching. Thus, Tutte's theorem can be derived as a corollary of the Tutte–Berge formula, and the formula can be seen as a generalization of Tutte's theorem.
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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.