Secure Total Domination Number in Maximal Outerplanar Graphs

A subset S of vertices in a graph G is a secure total dominating set of G if S is a total 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 total dominating set of G. We show that if G is a maximal outerplanar graph of order n, then G has a total secure dominating set of size at most 2n/3. Moreover, if an outerplanar graph G of order n, then each secure total dominating set has at least (n+2) /3 vertices. We show that these bounds are best possible.