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Is there a perfect matching or a maximum cardinality matching?


Asked by Marlowe Cisneros on Dec 09, 2021 FAQ



Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs: In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched.
Furthermore,
Maximum Cardinality Matching (MCM) problem is a Graph Matching problem where we seek a matching M that contains the largest possible number of edges. A possible variant is Perfect Matching where all V vertices are matched, i.e. the cardinality of M is V/2.
One may also ask, A maximum matching is a matching of maximum size (maximum number of edges). In a maximum matching, if any edge is added to it, it is no longer a matching.
Just so,
A perfect matching (a.k.a. 1-factor) is a matching which matches all vertices of the graph. That is, every vertex of the graph is incident to exactly one edge of the matching. Every perfect matching is maximum and hence maximal.
Thereof,
Maximum bipartite matching matches two elements with a property that no two edges share a vertex. Maximum matching is also called as maximum cardinality matching. Maximum matching is also called as maximum cardinality matching (i.e.) matching with the largest number of edges.