Paired-domination in generalized claw-free graphs

Abstract

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998) 199–206). 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 paired-domination number of G, denoted by \(\gamma_{\rm pr}(G)\), is the minimum cardinality of a paired-dominating set of G. If G does not contain a graph F as an induced subgraph, then G is said to be F-free. Haynes and Slater (Networks 32 (1998) 199–206) showed that if G is a connected graph of order \(n \ge 3\), then \(\gamma_{\rm pr}(G) \le n-1\) and this bound is sharp for graphs of arbitrarily large order. Every graph is \(K_{1,a+2}\)-free for some integer a ≥ 0. We show that for every integer a ≥ 0, if G is a connected \(K_{1,a+2}\)-free graph of order n ≥ 2, then \(\gamma_{\rm pr}(G) \le 2(an + 1)/(2a+1)\) with infinitely many extremal graphs.