Paired versus double domination in forbidden graph classes

A set D of vertices in a graph G is a dominating set of G if every vertex not in D has a neighbor in D, where two vertices are neighbors if they are adjacent. If the dominating set D of G has the additional property that the subgraph induced by D contains a perfect matching (not necessarily as an induced subgraph), then D is a paired dominating set of G. The paired domination number of G, denoted by γpr(G), is the minimum cardinality of a paired dominating set of G. A set D⊆V(G) is a double dominating set of G if every vertex in V(G)\D has at least two neighbors in D, and every vertex in D has a neighbor in D. The double domination number of G, denoted by γ×2(G), is the minimum cardinality of a double dominating set of G. Chellali and Haynes (Util Math 67:161–171, (2005)) showed that if G is a claw-free graph without isolated vertices, then the paired domination number of G is at most the double domination number of G. In this paper, we show that if G is a H-free graph for some H∈{P5,2K2∪K1,fork} without isolated vertices, then γpr(G)≤γ×2(G).