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 . be a 2-connected K 1 , 5 -free 3-vertex-critical graph of odd order. For any vertex v ∈ V ( G ) , we show that G − v has a perfect matching (except two graphs), which solves a conjecture posed by Ananchuen and Plummer [N. Ananchuen, M.D. Plummer, Matchings in 3-vertex critical graphs: The odd case, Discrete Math., 307 (2007) 1651–1658].