Abstract
Let \(G=(V,E)\) be a graph and \(p\) be a positive integer. A subset \(S\subseteq V\) is called a \(p\)-dominating set if each vertex not in \(S\) has at least \(p\) neighbors in \(S\). The \(p\)-domination number \(\gamma _p(G)\) is the size of a smallest \(p\)-dominating set of \(G\). The \(p\)-reinforcement number \(r_p(G)\) is the smallest number of edges whose addition to \(G\) results in a graph \(G^{\prime }\) with \(\gamma _p(G^{\prime })< \gamma _p(G)\). In this paper, we give an original study on the \(p\)-reinforcement, determine \(r_p(G)\) for some graphs such as paths, cycles and complete \(t\)-partite graphs, and establish some upper bounds on \(r_p(G)\). In particular, we show that the decision problem on \(r_p(G)\) is NP-hard for a general graph \(G\) and a fixed integer \(p\ge 2\).
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Acknowledgments
The work was supported by NNSF of China (No.10711233) and the Fundamental Research Fund of NPU (No. JC201150)
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Lu, Y., Hu, FT. & Xu, JM. On the \(p\)-reinforcement and the complexity. J Comb Optim 29, 389–405 (2015). https://doi.org/10.1007/s10878-013-9597-9
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DOI: https://doi.org/10.1007/s10878-013-9597-9