A characterization of maximal non-k-factor-critical graphs

A graph G of order p is k-factor-critical,where p and k are positive integers with the same parity, if the deletion of any set of k vertices results in a graph with a perfect matching. G is called maximal non-k-factor-critical if G is not k-factor-critical but image is k-factor-critical for every missing edge image. A connected graph G with a perfect matching on image vertices is k-extendable, for image, if for every matching M of size k in G there is a perfect matching in G containing all edges of M. G is called maximal non-k-extendable if G is not k-extendable but image is k-extendable for every missing edge image . A connected bipartite graph G with a bipartitioning set image such that image is maximal non-k-extendable bipartite if G is not k-extendable but image is k-extendable for any edge image with image and image. A complete characterization of maximal non-k-factor-critical graphs, maximal non-k-extendable graphs and maximal non-k-extendable bipartite graphs is given.