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The bondage number $b(G)$ of a graph $G$ is the smallest number of edgeswhose 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 aspossible, and assume $\chi\leq0$. Gagarin-Zverovich and Huang have recentlyfound upper bounds of $b(G)$ in terms of the maximum degree $\Delta(G)$ and theEuler characteristic $\chi(G)=\chi$. In this paper we prove a better upperbound $b(G)\leq\Delta(G)+\lfloor t\rfloor$ where $t$ is the largest real rootof the cubic equation $z^3 + z^2 + (3\chi - 8)z + 9\chi - 12=0$; this upperbound 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 withits Euler characteristic $\chi$.

New upper bounds for the bondage number of a graph in terms of its maximum degree and Euler characteristic