Abstract
A connected graph G has property E(m, n) (or more briefly "G is E(m, n)") if for every pair of disjoint matchings M and N in G with |M|=m and |N|=n respectively, there is a perfect matching F in G such that M subset of F and N boolean AND F= null . In particular, a graph which has the E(m, 0) property is said to be m-extendable. Also a graph G which has the property that G-u-v has a perfect matching for every pair of distinct vertices u and v is said to be bicritical. In the present paper we investigate further the interrelation between extendability and criticality. In Lovasz and Plummer (Matching theory, North-Holland Publisher, Amsterdam, 1986) author proved that a 2-extendable graph is either 1-extendable and bipartite, or else bicritical (clearly a graph cannot be both). In the present paper we generalize this result by extending the notion of bicriticality in several ways and studying the relationships of these extensions to extendability.