Bounds on the semipaired domination number of graphs with minimum degree at least two

Let G be a graph with vertex set V and no isolated vertices. A subset $$S \subseteq V$$ S ⊆ V is a semipaired dominating set of G if every vertex in $$V {\setminus } S$$ 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 $$\gamma _\mathrm{pr2}(G)$$ γ pr 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 $$\gamma _\mathrm{pr2}(G) \le \frac{1}{2}(n+1)$$ γ pr 2 ( G ) ≤ 1 2 ( n + 1 ) . Further, we show that if $$n \not \equiv 3 \, (\mathrm{mod}\, 4)$$ n ≢ 3 ( mod 4 ) , then $$\gamma _\mathrm{pr2}(G) \le \frac{1}{2}n$$ γ pr 2 ( G ) ≤ 1 2 n , and we show that for every value of $$n \equiv 3 \, (\mathrm{mod}\, 4)$$ n ≡ 3 ( mod 4 ) , there exists a connected graph G of order n with minimum degree at least 2 satisfying $$\gamma _\mathrm{pr2}(G) = \frac{1}{2}(n+1)$$ γ pr 2 ( G ) = 1 2 ( n + 1 ) . As a consequence of this result, we prove that every graph G of order n with minimum degree at least 3 satisfies $$\gamma _\mathrm{pr2}(G) \le \frac{1}{2}n$$ γ pr 2 ( G ) ≤ 1 2 n .