Paired-domination in claw-free graphs with minimum degree at least three

Abstract

Let G=(V,E) be a simple graph without isolated vertices. A set S⊆V is called a paired-dominating set if every vertex in V∖S has at least one neighbor in S and the subgraph induced by S contains 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 show that γpr(G)≤4n∕7 if G is a claw-free graph of order n with minimum degree at least three. The statement partly confirms the conjecture proposed by Goddard and Henning in 2009.