On the secure domination numbers of maximal outerplanar graphs

A subset S of vertices in a graph G is a secure dominating set of G if S is a dominating set of G and, for each vertex uS, there is a vertex vS such that uv is an edge and (S{v}){u} is also a dominating set of G. We show that if G is a maximal outerplane graph of n vertices, then G has a secure dominating set of size at most 3n7. Moreover, if a maximal outerplane graph G has no internal triangles, it has a secure dominating set of size at most n3. Finally, we show that any secure dominating set of a maximal outerplane graph without internal triangles has more than n4 vertices.