The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhood covers all the vertices of the graph. The packing number $\rho(G)$ of $G$ is the maximum size of a vertex set whose closed neighborhoods are pairwise disjoint. In this paper we study graph classes ${\cal G}$ such that $\gamma(G)/\rho(G)$ is bounded by a constant $c_{\cal G}$ for each $G\in {\cal G}$. We propose an inductive proof technique to prove that if $\cal G$ is the class of $2$-degenerate graphs, then there is such a constant bound $c_{\cal G}$. We note that this is the first monotone, dense graph class that is shown to have constant ratio. We also show that the classes of AT-free and unit-disk graphs have bounded ratio. In addition, our technique gives improved bounds on $c_{\cal G}$ for planar graphs, graphs of bounded treewidth or bounded twin-width. Finally, we provide some new examples of graph classes where the ratio is unbounded.

翻译：图G的支配数$\gamma(G)$是指其闭邻域覆盖图所有顶点的最小顶点集大小。图G的包装数$\rho(G)$是指其闭邻域两两不交的最大顶点集大小。本文研究满足以下条件的图类${\cal G}$：对任意$G\in {\cal G}$，比值$\gamma(G)/\rho(G)$存在常数上界$c_{\cal G}$。我们提出了一种归纳证明方法，证明若$\cal G$为2-退化图类，则存在这样的常数界$c_{\cal G}$。值得注意的是，这是首个被证明具有常数比的单调稠密图类。我们还证明了AT-free图和单位圆盘图类具有有界比值。此外，我们的方法为平面图、有界树宽图及有界双宽图类中的$c_{\cal G}$提供了更优上界。最后，我们给出了一些比值无界的图类新实例。