# Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

# Matching

Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_589

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 V1 and V2 and every edge in Ehas...

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