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Upper bounds for the bondage number of graphs on topological surfaces. (English) Zbl 1277.05125

Summary: 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 \(\varDelta(G)\) and embeddable on an orientable surface of genus \(h\) and a non-orientable surface of genus \(k\), \[ b(G)\leq\min\{\varDelta(G)+h+2,\varDelta (G)+k+1\}. \] 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.

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C07 Vertex degrees
05C35 Extremal problems in graph theory
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References:

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