On critically k-extendable graphs

ON CRITICALLY k-EXTENDABLE GRAPHS N. Anunchuen and L. Caccetta School of Mathematics and Statistics Curtin University of Technology GPO Box U1987 Perth 6001 Western Australia Let G be a simple connected graph on 2n vertices with a perfect matching. G is k-extendable if for any set M of k independent edges, there exists a perfect matching in G containing all the edges of M. G is critically k-extendable if G is k-extendable but G + uv is not k-extendable for any non-adjacent pair of vertices u and v of G. The problem that arises is that of characterizing k-extendable and cri tically k-extendable graphs. This problem has been studied for k-extendable graphs and a number of results have been obtained. In particular, complete characterizations have been obtained for the case k = 1. Critically k-extendable graphs have not been studied. In this paper, we focus on the problem of characterizing cri tically k-extendable graphs. Complete characterizations are presented for k = 1, n 2, n 1 and n.