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.