Graphs with large semipaired domination number

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. We show that if G is a connected graph G of order n ≥ 3, then γ_pr2(G) ≤ ₃ ²n, and we characterize the extremal graphs achieving equality in the bound.