Paired versus double domination in K _1,r -free graphs

A vertex in G is said to dominate itself and its neighbors. A subset S of vertices is a dominating set if S dominates every vertex of G. A paired-dominating set is a dominating set whose induced subgraph contains a perfect matching. The paired-domination number of a graph G, denoted by γ _pr(G), is the minimum cardinality of a paired-dominating set in G. A subset S⊆V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ _×2(G). A claw-free graph is a graph that does not contain K _1,3 as an induced subgraph. Chellali and Haynes (Util. Math. 67:161-171, 2005) showed that for every claw-free graph G, we have γ _pr(G)≤ γ×2 (G). In this paper we extend this result by showing that for r≥2, if G is a connected graph that does not contain K _1,r as an induced subgraph, then γ_pr(G) ≤ (2r²-6r+6/r(r-1) γ _×2(G).