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$\gamma(G)$表示~$G$中控制集的最小基数。与控制数对偶的概念是图的打包数。图~$G$的打包是指顶点之间两两距离至少为3的顶点集，打包数$\rho(G)$表示此类集合的最大基数。众所周知，不等式$\rho(G) \leq \gamma(G)$成立。Henning等人猜想：若$G$为亚三次图，则$\gamma(G) \leq 2\rho(G)+1$。本文通过证明以下结果推进该猜想：若$G$为二分三次图，则$\gamma(G) \leq \frac{120}{49}\rho(G)$；若$G$为最大外平面图，则$\gamma(G) \leq 3\rho(G)$；若$G$为双凸图，则$\gamma(G) \leq 2\rho(G)$。此外，在最后一种情形中，我们证明该上界是紧的。