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The bondage numbers of graphs with small crossing numbers. (English) Zbl 1118.05073

Summary: The bondage number \(b(G)\) of a nonempty graph \(G\) is the cardinality of a smallest edge set whose removal from \(G\) results in a graph with domination number greater than the domination number \(\gamma (G)\) of \(G\). L. Kang and J. Yuan [Discrete Math. 222, No. 1–3, 191–198 (2000; Zbl 0961.05055)] proved \(b(G)\leq 8\) for every connected planar graph \(G\). M. Fischermann, D. Rautenbach and L. Volkmann [Discrete Math. 260, No. 1–3, 57–67 (2003; Zbl 1009.05104)] obtained some further results for connected planar graphs. We generalize their results to connected graphs with small crossing numbers.

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C10 Planar graphs; geometric and topological aspects of graph theory
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References:

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