# Matching

**DOI:**https://doi.org/10.1007/1-4020-0611-X_589

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Matching problems form an important branch of *graphtheory*. They are of particular interest because of their application to problems found in Operations Research. Matching problems also form a class of integer linear programming problems which can be solved in polynomial time. A good description of the historical development of matching problems and their solutions is contained in the preface of Lovasz and Plummer (1986).

Given a simple non-directed graph *G* = [*V*, *E*] (where *E* is a set of edges), then a *matching* is defined as a subset of edges *M* such that no two edges of *M* are adjacent. A matching is said to *span* a set of vertices *X* in *G* if every vertex in *X* is incident with an edge of the matching. A *perfect matching* is a matching which spans *V*. A *maximum matching* is a matching of maximum cardinality, i.e. a matching with the maximum number of members in the set.

A graph is called a *bipartite graph* if the set of vertices *V* is the disjoint union of sets *V*_{1} and *V*_{2} and every edge in *E*has...

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