A note on the double domination number in maximal outerplanar and planar graphs

Abstract

In a graph, a vertex dominates itself and its neighbors. A subset S of vertices of a graph G is a double dominating set of G if S dominates every vertex of G at least twice. The double domination number γ×2(G) of G is the minimum cardinality of a double dominating set of G. In this paper, we prove that the double domination number of a maximal outerplanar graph G of order n is bounded above by n+k/2, where k is the number of pairs of consecutive vertices of degree two and with distance at least 3 on the outer cycle. We also prove that γ×2(G) ≤ 5n/8 for a Hamiltonian maximal planar graph G of order n ≥ 7.