Semipaired domination in maximal outerplanar graphs

Abstract

A subset S of vertices in a graph G is a dominating set if every vertex in $V(G) {\setminus } S$ is adjacent to a vertex in S. If the graph G has no isolated vertex, then a semipaired dominating set of G is a dominating set of G with the additional property that the set 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. Let G be a maximal outerplanar graph of order n with $n_2$ vertices of degree 2. We show that if $n \ge 5$, then $\gamma _{\mathrm{pr2}}(G) \le \frac{2}{5}n$. Further, we show that if $n \ge 3$, then $\gamma _{\mathrm{pr2}}(G) \le \frac{1}{3}(n+n_2)$. Both bounds are shown to be tight.