Matching extension in K1, r-free graphs with independent claw centers

We say that a graph G is k-extendable if every set of k independent edges of G can be extended to a perfect matching. In the paper it is proved that if G is an even (2k + 1)-connected K 1;k+3-free graph such that the set of all centers of claws is independent, then G is k-extendable. As a corollary we obtain an analogous result for almost claw-free graphs and for claw-free graphs, thus extending a result by M.D. Plummer.