  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. Let
$G$ be embeddable on a surface whose Euler characteristic $\chi$ is as large as
possible, and assume $\chi\leq0$. Gagarin-Zverovich and Huang have recently
found upper bounds of $b(G)$ in terms of the maximum degree $\Delta(G)$ and the
Euler characteristic $\chi(G)=\chi$. In this paper we prove a better upper
bound $b(G)\leq\Delta(G)+\lfloor t\rfloor$ where $t$ is the largest real root
of the cubic equation $z^3 + z^2 + (3\chi - 8)z + 9\chi - 12=0$; this upper
bound is asymptotically equivalent to $b(G)\leq\Delta(G)+1+\lfloor
\sqrt{4-3\chi} \rfloor$. We also establish further improved upper bounds for
$b(G)$ when the girth, order, or size of the graph $G$ is large compared with
its Euler characteristic $\chi$.
