Upper paired-domination in claw-free graphs

Abstract

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The maximum cardinality of a minimal paired-dominating set of G is the upper paired-domination number of G, denoted by Γpr(G). We establish bounds on Γpr(G) for connected claw-free graphs G in terms of the number n of vertices in G with given minimum degree ?. We show that Γpr(G)?4n/5 if ?=1 and n?3, Γpr(G)?3n/4 if ?=2 and n?6, and Γpr(G)?2n/3 if ??3. All these bounds are sharp. Further, if n?6 the graphs G achieving the bound Γpr(G)=4n/5 are characterized, while for n?9 the graphs G with ?=2 achieving the bound Γpr(G)=3n/4 are characterized.