Upper bounds for the bondage number of graphs on topological surfaces

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 $Δ(G)$ and embeddable on an orientable surface of genus h and a non-orientable surface of genus k, $b(G)\le \min\{Δ(G)+h+2, Δ(G)+k+1\}$. This generalizes known upper bounds for planar and toroidal graphs.

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