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The bondage number of a graph is the smallest number of its edges whoseremoval results in a graph having a larger domination number. We provideconstant upper bounds for the bondage number of graphs on topological surfaces,improve upper bounds for the bondage number in terms of the maximum vertexdegree and the orientable and non-orientable genera of the graph, and showtight lower bounds for the number of vertices of graphs 2-cell embeddable ontopological surfaces of a given genus. Also, we provide stronger upper boundsfor graphs with no triangles and graphs with the number of vertices larger thana certain threshold in terms of the graph genera. This settles Teschner'sConjecture in positive for almost all graphs.

The bondage number of graphs on topological surfaces and Teschner's conjecture