  The bondage number b(G) of a graph G is the smallest number of edges of G
whose removal from G results in a graph having the domination number larger
than that of G. We show that, for a graph G having the maximum vertex degree
$\Delta(G)$ and embeddable on an orientable surface of genus h and a
non-orientable surface of genus k, $b(G)\le \min\{\Delta(G)+h+2,
\Delta(G)+k+1\}$. This generalizes known upper bounds for planar and toroidal
graphs.
