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~$\rho(G) \leq \gamma(G)$ is well-known. Henning et al.\ conjectured that~$\gamma(G) \leq 2\rho(G)+1$ if~$G$ is subcubic. In this paper, we progress towards this conjecture by showing that~${\gamma(G) \leq \frac{120}{49}\rho(G)}$ if~$G$ is a bipartite cubic graph. We also show that $\gamma(G) \leq 3\rho(G)$ if~$G$ is a maximal outerplanar graph, and that~$\gamma(G) \leq 2\rho(G)$ if~$G$ is a biconvex graph. Moreover, in the last case, we show that this upper bound is tight.

翻译：给定图~$G$，控制数~$\gamma(G)$ 定义为~$G$ 中最小控制集的基数。与控制数对偶的概念是图的打包数。图~$G$ 的打包是顶点集合，其中任意两个顶点的距离至少为3。打包数~$\rho(G)$ 是此类集合的最大基数。此外，不等式~$\rho(G) \leq \gamma(G)$ 是众所周知的。Henning 等人猜想：若~$G$ 为次三次图，则~$\gamma(G) \leq 2\rho(G)+1$。本文中，我们向该猜想推进，证明了当~$G$ 为二分三次图时，有~${\gamma(G) \leq \frac{120}{49}\rho(G)}$。此外，我们还证明了对于极大外平面图~$G$，有 $\gamma(G) \leq 3\rho(G)$；对于双凸图~$G$，有~$\gamma(G) \leq 2\rho(G)$，并且在后一情形中，我们证明该上界是紧的。