Semipaired domination in maximal outerplanar graphs

A subset S of vertices in a graph G is a dominating set if every vertex in $$V(G) {\setminus } S$$ V ( G ) \ 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 $$\gamma _{\mathrm{pr2}}(G)$$ γ pr 2 ( G ) is the minimum cardinality of a semipaired dominating set of G. Let G be a maximal outerplanar graph of order n with $$n_2$$ n 2 vertices of degree 2. We show that if $$n \ge 5$$ n ≥ 5 , then $$\gamma _{\mathrm{pr2}}(G) \le \frac{2}{5}n$$ γ pr 2 ( G ) ≤ 2 5 n . Further, we show that if $$n \ge 3$$ n ≥ 3 , then $$\gamma _{\mathrm{pr2}}(G) \le \frac{1}{3}(n+n_2)$$ γ pr 2 ( G ) ≤ 1 3 ( n + n 2 ) . Both bounds are shown to be tight.