Paired-domination number of a graph and its complement

Abstract

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 γ p r ( 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 γ p r ( G ) > 4 and γ p r ( G ¯ ) > 4 , then γ p r ( G ) + γ p r ( G ¯ ) ≤ 3 + min { δ ( G ) , δ ( G ¯ ) } . –if δ ( G ) ≥ 2 and δ ( G ¯ ) ≥ 2 , then γ p r ( G ) + γ p r ( G ¯ ) ≤ 2 n 3 + 4 and γ p r ( G ) + γ p r ( G ¯ ) ≤ 2 n 3 + 2 if moreover n ≥ 21 .