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 ? 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 ? pr -critical, if for any vertex v of G that is not adjacent to any vertex of degree one, ? pr (G?v)<? pr (G). A ? pr -critical graph G is said to be k-? pr -critical if ? pr (G)=k. In this paper, we firstly show that every 4-? pr -critical graph of even order has a perfect matching if it is K 1,5-free and every 4-? pr -critical graph of odd order is factor-critical if it is K 1,5-free. Secondly, we show that every 6-? pr -critical graph of even order has a perfect matching if it is K 1,4-free.