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The p-Bondage Number of Trees

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Abstract

Let p be a positive integer and G = (V, E) be a simple graph. A p-dominating set of G is a subset \({D\,{\subseteq}\, V}\) such that every vertex not in D has at least p neighbors in D. The p-domination number of G is the minimum cardinality of a p-dominating set of G. The p-bondage number of a graph G with (ΔG) ≥ p is the minimum cardinality among all sets of edges \({B\subseteq E}\) for which γ p (GB) > γ p (G). For any integer p ≥ 2 and tree T with (ΔT) ≥ p, this paper shows that 1 ≤  b p (T) ≤ (ΔT) − p + 1, and characterizes all trees achieving the equalities.

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Correspondence to Jun-Ming Xu.

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The work was supported by NNSF of China (No. 10671191, 10701068).

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Lu, Y., Xu, JM. The p-Bondage Number of Trees. Graphs and Combinatorics 27, 129–141 (2011). https://doi.org/10.1007/s00373-010-0956-3

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  • DOI: https://doi.org/10.1007/s00373-010-0956-3

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