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 u⁄∈S, there is a vertex v∈S 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 ⌈3n∕7⌉. Moreover, if a maximal outerplane graph G has no internal triangles, it has a secure dominating set of size at most ⌈n∕3⌉. Finally, we show that any secure dominating set of a maximal outerplane graph without internal triangles has more than n∕4 vertices.