On dominating sets of maximal outerplanar and planar graphs

A set D⊆V(G) of a graph G is a dominating set if every vertex v∈V(G) is either in D or adjacent to a vertex in D. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set of G. Campos and Wakabayashi (2013) and Tokunaga (2013) proved independently that if G is an n-vertex maximal outerplanar graph having t vertices of degree 2, then γ(G)≤n+t4. We improve their upper bound by showing γ(G)≤n+k4, where k is the number of pairs of consecutive 2-degree vertices with distance at least 3 on the outer cycle. Moreover, we prove that γ(G)≤5n16 for a Hamiltonian maximal planar graph G with n≥7 vertices.