Domination and packing in graphs

Abstract

Given a graph~$G$, the domination number, denoted by~$\gamma (G)$, is the minimum cardinality of a dominating set in~$G$. Dual to the notion of domination number is the packing number of a graph. A packing of~$G$ is a set of vertices whose pairwise distance is at least three. The packing number~$\rho (G)$ of~$G$ is the maximum cardinality of one such set. Furthermore, the inequality~$\rho (G)\le \gamma (G)$ is well-known. Henning et al.\ conjectured that~$\gamma (G)\le 2\rho (G)+1$ if~$G$ is subcubic. In this paper, we progress towards this conjecture by showing that~$\gamma (G)\le \frac{120}{49}\rho (G)$ if~$G$ is a bipartite cubic graph. We also show that $\gamma (G)\le 3\rho (G)$ if~$G$ is a maximal outerplanar graph, and that~$\gamma (G)\le 2\rho (G)$ if~$G$ is a biconvex graph. Moreover, in the last case, we show that this upper bound is tight.

12 pages, 6 figures