Paired Domination Vertex Critical Graphs

Let γ _pr (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, γ _pr (G – v) < γ _pr (G). We call these graphs γ _pr -critical. In this paper, we present a method of constructing γ _pr -critical graphs from smaller ones. Moreover, we show that the diameter of a γ _pr -critical graph is at most 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]. Xinmin Hou: Research supported by NNSF of China (No.10701068 and No.10671191).