Perfect matchings in paired domination vertex critical graphs

A vertex subset S of a graph G=(V,E) is a paired dominating set if every vertex of G is adjacent to some vertex in S and the subgraph induced by S contains a perfect matching. The paired domination number of G, denoted by gamma (pr) (G), is the minimum cardinality of a paired dominating set of G. A graph with no isolated vertex is called paired domination vertex critical, or briefly gamma (pr) -critical, if for any vertex v of G that is not adjacent to any vertex of degree one, gamma (pr) (G-v)<gamma (pr) (G). A gamma (pr) -critical graph G is said to be k-gamma (pr) -critical if gamma (pr) (G)=k. In this paper, we firstly show that every 4-gamma (pr) -critical graph of even order has a perfect matching if it is K (1,5)-free and every 4-gamma (pr) -critical graph of odd order is factor-critical if it is K (1,5)-free. Secondly, we show that every 6-gamma (pr) -critical graph of even order has a perfect matching if it is K (1,4)-free.