On the connectivity and matchings in 3-vertex-critical claw-free graphs

Abstract

A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. The cardinality of any smallest dominating set in G is denoted by γ(G) and called the domination number of G. Graph G is said to be γ-vertex-critical if γ(G -v) < γ(G), for every vertex v in G. For |V(G)| ≡ k (mod 2), graph G is said to be k-factor-critical if G - S has a perfect matching for every subset S ⊆ V(G) with |S| = k. In two previous papers, (cf. [AP1, AP2]), the study of matchings in 3-vertex-critical graphs was begun. In the present paper, results about connectivity and k-factor-criticality are presented, for the case in which the 3-vertex-critical graphs are also claw-free.