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. In a sense, the bondage number b(G) measures integrity and reliability of the domination number γ(G) with respect to the edge removal, which corresponds, e.g., to link failures in communication networks. We show that, for a graph G having the maximum vertex degree ∆(G) and embeddable on an orientable surface of genus h and a non-orientable surface of genus k,