Domination and packing in graphs

The dominating number $\gamma (G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhoods cover all vertices of $G$, while the packing number $\rho (G)$ is the maximum size of a vertex set whose closed neighborhoods are pairwise disjoint. In this paper we investigate graph classes $\mathcal{G}$ for which the ratio $\gamma (G)\mathrm{/}\rho (G)$ is bounded by a constant $c_{\mathcal{G}}$ for every $G\in \mathcal{G}$. Our main result is an improved upper bound on this ratio for planar graphs. We also extend the list of graph classes admitting a bounded ratio by showing this for chordal bipartite graphs and for homogeneously orderable graphs. In addition, we provide a simple, direct proof for trees.