  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.
