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 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)≤min{Δ(G)+h+2,Δ(G)+k+1}. This generalizes known upper bounds for planar and toroidal graphs, and can be improved for bigger values of the genera h and k by adjusting the proofs.