The Paired-domination and the Upper Paired-domination Numbers of Graphs
- Włodzimierz Ulatowski, Dalibor Fronček
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 γpr(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 ≥ 3, then γpr(G) ≤ n−1 and this bound is sharp for graphs of arbitrarily large order. Every graph is K1,a+2-free for some integer a ≥ 0. We show that for every integer a ≥ 0, if G is a connected K1,a+2-free graph of order n ≥ 2, then γpr(G) ≤ 2(an + 1)/(2a + 1) with infinitely many extremal graphs.