On two equimatchable graph classes Original Research Article

A graph G is said to be equimatchable if every matching in G extends to (i.e., is a subset of) a maximum matching. In this paper, we use the Gallai–Edmonds decomposition theory for matchings to determine the equimatchable members of two important graph classes. We find that there are precisely 23 3-connected planar graphs (i.e., 3-polytopes) which are equimatchable and that there are only two cubic equimatchable graphs.