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

Abstract

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