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 γpr 2(G) is the minimum cardinality of a semipaired dominating set of G. We show that if G is a connected graph of order n with minimum degree at least 2, then γpr2(G)≤12(n+1). Further, we show that if n≢3(mod4), then γpr2(G)≤12n, and we show that for every value of n≡3(mod4), there exists a connected graph G of order n with minimum degree at least 2 satisfying γpr2(G)=12(n+1). As a consequence of this result, we prove that every graph G of order n with minimum degree at least 3 satisfies γpr2(G)≤12n.

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. We present a method of building trees having a unique minimum semipaired dominating set.

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) ≤ 32 n, and we characterize the extremal graphs achieving equality in the bound.

Let G be a graph with vertex set V. 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 is the minimum cardinality of a semipaired dominating set of G. We characterize the trees having a unique minimum semipaired dominating set. We also determine an upper bound on the semipaired domination number of these trees and characterize the trees attaining this bound.

Let G be a graph with vertex set V and no isolated vertices, and let S be a dominating set of V. The set S is a semitotal dominating set of G if every vertex in S is within distance 2 of another vertex of S. And, S is a semipaired dominating set of G if S can be partitioned into 2-element subsets such that the vertices in each 2-set are at most distance two apart. The semitotal domination number γt 2(G) is the minimum cardinality of a semitotal dominating set of G, and the semipaired domination number γpr 2(G) is the minimum cardinality of a semipaired dominating set of G. For a graph without isolated vertices, the domination number γ(G) , the total domination γt(G) , and the paired domination number γpr(G) are related to the semitotal and semipaired domination numbers by the following inequalities: γ(G) ≤ γt 2(G) ≤ γt(G) ≤ γpr(G) and γ(G) ≤ γt 2(G) ≤ γpr 2(G) ≤ γpr(G) ≤ 2 γ(G). Given two graph parameters μ and ψ related by a simple inequality μ(G) ≤ ψ(G) for every graph G having no isolated vertices, a graph is (μ, ψ) -perfect if every induced subgraph H with no isolated vertices satisfies μ(H) = ψ(H). Alvarado et al. (Discrete Math 338:1424–1431, 2015) consider classes of (μ, ψ) -perfect graphs, where μ and ψ are domination parameters including γ, γt and γpr. We study classes of perfect graphs for the possible combinations of parameters in the inequalities when γt 2 and γpr 2 are included in the mix. Our results are characterizations of several such classes in terms of their minimal forbidden induced subgraphs.

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 γPr2(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) ≤ γPr2(G) ≤ 2γ(G) and 2/3γt(G) ≤ γPr2(T) ≤ γ 4/3γt(G), where γ(G) and γt(G) denote the domination and total domination numbers of G. We characterize the trees G for which γPr2(G) = 2γ(G).