Paired-Domination in Claw-Free Cubic 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 minimum cardinality of a paired-dominating set of G is the paired-domination number of G, denoted by γ_pr(G). If G does not contain a graph F as an induced subgraph, then G is said to be F-free. In particular if F=K_1,3 or K₄−e, then we say that G is claw-free or diamond-free, respectively. Let G be a connected cubic graph of order n. We show that (i) if G is (K_1,3,K₄−e,C₄)-free, then γ_pr(G)≤3n/8; (ii) if G is claw-free and diamond-free, then γ_pr(G)≤2n/5; (iii) if G is claw-free, then γ_pr(G)≤n/2. In all three cases, the extremal graphs are characterized.