  The bondage number $b(G)$ of a graph $G$ is the smallest number of edges
whose removal from $G$ results in a graph with larger domination number. An
orientable surface $\mathbb{S}_h$ of genus $h$, $h \geq 0$, is obtained from
the sphere $\mathbb{S}_0$ by adding $h$ handles. A non-orientable surface
$\mathbb{N}_q$ of genus $q$, $q \geq 1$, is obtained from the sphere by adding
$q$ crosscaps. The Euler characteristic of a surface is defined by
$\chi(\mathbb{S}_h) = 2 - 2h$ and $\chi(\mathbb{S}_q)= 2-q$. Let $G$ be a
connected graph of order $n$ which is 2-cell embedded on a surface $\mathbb{M}$
with $\chi(\mathbb{M})= \chi$. We prove that $b(G) \leq 7+i$ when $\mathbb{M} =
\mathbb{N}_i$, $i=1,2,3$, and $b(G) \leq 12$ when $\mathbb{M} \in
\{\mathbb{N}_4, \mathbb{S}_2\}$. We give new arguments that improve the known
upper bounds on the bondage number at least when $-7\chi/(\delta(G) - 5) < n
\leq -12\chi$, $\delta(G) \geq 6$, where $\delta(G)$ is the minimum degree of
$G$. We obtain sufficient conditions for the validity of the inequality $b(G)
\leq 2s-2$, provided $G$ has degree $s$ vertices. In particular, we prove that
if $\delta (G) = \delta \geq 6$, $\chi \leq -1$ and $-14\chi < \delta - 4 +
2(\delta -5)n$ then $b(G) \leq 2\delta -2$. We show that if $\gamma (G) =
\gamma \not = 2$, where $\gamma (G)$ is the domination number of $G$, then $n
\geq \gamma + (1 + \sqrt{9+8\gamma-8\chi})/2$; the bound is tight. We also
present upper bounds for the bondage number of graphs in terms of the girth,
domination number and Euler characteristic. As a corollary we prove that if
$\gamma(G) \geq 4$ and $\chi \leq -1$, then $b(G) \leq 11 - 24\chi/(9 +
\sqrt{41 - 8\chi})$. Several unanswered questions are posed.
