Paired-domination number of a graph and its complement

A paired-dominating set of a graph G = (V, E) with no isolated vertex is a dominating set of vertices inducing a graph with a perfect matching. The paired-domination number of G, denoted by γpr (G), is the minimum cardinality of a paired-dominating set of G. We consider graphs of order n ≥ 6, minimum degree δ such that G and G do not have an isolated vertex and we prove that – if γpr (G) > 4 and γpr (G) > 4, then γpr (G)+ γpr (G) ≤ 3+min{δ(G), δ(G)}. – if δ(G) ≥ 2 and δ(G) ≥ 2, then γpr (G)+ γpr (G) ≤ 2n 3 + 4 and γpr (G)+ γpr (G) ≤ 2n 3 + 2 if moreover n ≥ 21. c © 2007 Elsevier B.V. All rights reserved.