Semipaired domination in graphs

In honor of Professor Peter Slater's work on paired domination, we introduce a relaxed version of paired domination, namely semipaired domination. Let G be a graph with vertex set V and no isolated vertices. A subset S ⊆ V is a semipaired dominating set of G if every vertex in V \ S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number γ_Pr2(G) is the minimum cardinality of a semipaired dominating set of G. In this paper, we study the semipaired domination versus other domination parameters. For example, we show that γ(G) ≤ γ_Pr2(G) ≤ 2γ(G) and 2/3γ_t(G) ≤ γ_Pr2(T) ≤ γ 4/3γ_t(G), where γ(G) and γ_t(G) denote the domination and total domination numbers of G. We characterize the trees G for which γ_Pr2(G) = _2γ(G).