Extendability and factor-criticality Original Research Article

A graph G of order n is p-factor-critical, where p is an integer of the same parity as n, if the removal of any set of p vertices of G results in a graph with a perfect matching. When n is even, G is p-extendable if any set of p independent edges is contained in a perfect matching. We show that for p even, every non-bipartite p-extendable graph is p-factor-critical, and every non-bipartite (p+1)-extendable graph G is such that G−e is p-factor-critical for every edge e of G. This generalizes a result of Plummer on bicritical graphs. As a consequence, we get an upper bound on the independence number of a non-bipartite p-extendable graph.