The enumeration of near-perfect matchings of factor-critical graphs

A matching of a graph is a near-perfect matching if it covers all but one vertex. A connected graph G is said to be factor-critical if G−v has perfect matchings for every vertex v of G. In this paper, the enumeration problems for near-perfect matchings in special factor-critical graphs are solved. From this, the exact values of the number of perfect matchings in special bicritical graphs are obtained.