m at h . C O ] 2 8 A ug 2 00 6 Factor - Critical Property in 3 - Dominating - Critical Graphs ∗

A vertex subset S of a graph G is a dominating set if every vertex of G either belongs to S or is adjacent to a vertex of S. The cardinality of a smallest dominating set is called the dominating number of G and is denoted by γ(G). A graph G is said to be γ-vertex-critical if γ(G− v) < γ(G), for every vertex v in G. Let G be a 2-connected K1,5-free 3-vertex-critical graph. For any vertex v ∈ V (G), we show that G − v has a perfect matching (except two graphs), which is a conjecture posed by Ananchuen and Plummer [2].