Upper paired domination in claw-free graphs
Résumé
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 δ. We show that Γpr (G) ≤ 4n/5 if δ = 1 and n ≥ 3, Γpr (G) ≤ 3n/4 if δ = 2 and n ≥ 6, and Γpr (G) ≤ 2n/3 if δ ≥ 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 δ = 2 achieving the bound Γpr (G) = 3n/4 are characterized.