Perfect matchings in paired domination vertex critical graphs

Abstract

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 γ _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 γ _pr -critical, if for any vertex v of G that is not adjacent to any vertex of degree one, γ _pr (G−v)<γ _pr (G). A γ _pr -critical graph G is said to be k-γ _pr -critical if γ _pr (G)=k. In this paper, we firstly show that every 4-γ _pr -critical graph of even order has a perfect matching if it is K _1,5-free and every 4-γ _pr -critical graph of odd order is factor-critical if it is K _1,5-free. Secondly, we show that every 6-γ _pr -critical graph of even order has a perfect matching if it is K _1,4-free.