Upper paired-domination in claw-free graphs

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 pr(G). We establish bounds on pr(G) for connected claw-free graphs G in terms of the number n of vertices in G with given minimum degree δ. We show that pr(G) ≤ 4n/5 if δ = 1 and n ≥ 3, pr(G) ≤ 3n/4 if δ = 2 and n ≥ 6, and pr(G) ≤ 2n/3 if δ ≥ 3. All these bounds are sharp. Further, if n≥ 6 the graphs G achieving the bound pr(G)= 4n/5 are characterized, while for n≥ 9 the graphs G with δ = 2 achieving the bound pr(G)= 3n/4 are characterized.