Paired Domination Vertex Critical Graphs

Abstract

Let γpr (G) denote the paired domination number of graph G. A graph G with no isolated vertex is paired domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γpr (G – v) < γpr (G). We call these graphs γpr -critical. In this paper, we present a method of constructing γpr -critical graphs from smaller ones. Moreover, we show that the diameter of a γpr -critical graph is at most $$\frac{3}{2}(\gamma_{pr} (G)-2)$$and the upper bound is sharp, which answers a question proposed by Henning and Mynhardt [The diameter of paired-domination vertex critical graphs, Czechoslovak Math. J., to appear].