Abstract
For a graph property P and a graph G, we define the domination subdivision number with respect to the property P to be the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to change the domination number with respect to the property P. In this paper we obtain upper bounds in terms of maximum degree and orientable/non-orientable genus for the domination subdivision number with respect to an induced-hereditary property, total domination subdivision number, bondage number with respect to an induced-hereditary property, and Roman bondage number of a graph on topological surfaces.
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