Double domination in maximal outerplanar graphs

Abstract

Abstract In graph G G , a vertex dominates itself and its neighbors. A subset S ⊆ V ( G ) S\subseteq V\left(G) is said to be a double-dominating set of G G if S S dominates every vertex of G G at least twice. The double domination number γ × 2 ( G ) {\gamma }_{\times 2}\left(G) is the minimum cardinality of a double dominating set of G G . We show that if G G is a maximal outerplanar graph on n ≥ 3 n\ge 3 vertices, then γ × 2 ( G ) ≤ 2 n 3 {\gamma }_{\times 2}\left(G)\le ⌊\frac{2n}{3}⌋ . Further, if n ≥ 4 n\ge 4 , then γ × 2 ( G ) ≤ min n + t 2 , n − t {\gamma }_{\times 2}\left(G)\le \min \left\{⌊\frac{n+t}{2}⌋,n-t\right\} , where t t is the number of vertices of degree 2 in G G . These bounds are shown to be tight. In addition, we also study the case that G G is a striped maximal outerplanar graph.