Domination and packing in graphs

Abstract

Given a graph G, the domination number $\gamma \left(G\right)$ is the minimum cardinality of a dominating set in G, and the packing number $\rho \left(G\right)$ is the minimum cardinality of a set of vertices whose pairwise distance is at least three. The inequality $\rho \left(G\right)\le \gamma \left(G\right)$ is well-known. Furthermore, Henning et al. conjectured that $\gamma \left(G\right)\le 2\rho \left(G\right)+1$ if G is subcubic. In this paper, we show that $\gamma \left(G\right)\le \frac{120}{49}\rho \left(G\right)$ if G is a bipartite cubic graph. This result is obtained by showing that $\rho \left(G\right)\ge \frac{7}{48}\left|V\right(G\left)\right|$ for this class of graphs, which improves a previous bound given by Favaron. We also show that $\gamma \left(G\right)\le 3\rho \left(G\right)$ if G is a maximal outerplanar graph, and that $\gamma \left(G\right)\le 2\rho \left(G\right)$ if G is a biconvex graph, where the latter result is tight.