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Abstract

A vertex subset S of a graph G is a 2-dominating set of G if every vertex not in S is adjacent to two vertices of S. The 2-domination number \(\gamma _2(G)\) is the minimum cardinality of a 2-dominating set of G. The 2-reinforcement number \(r_2(G)\) is the smallest number of extra edges whose addition to G results in a graph \(G'\) with \(\gamma _2(G')< \gamma _2(G)\). Let T be a tree. It is showed by Lu, Hu, and Xu that \(r_2(T)\le 3\). In this paper, we will show that \(r_2(T)=3\) if and only if there is a 2-dominating set S of T such that T contains neither S-vulnerable vertices nor S-vulnerable paths.

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Acknowledgments

The work was supported by NNSF of China (No. 11201374) and the Fundamental Research Funds for the Central University (NO. 3102014JCQ01074).

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Correspondence to You Lu.

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Communicated by Xueliang Li.

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Lu, Y., Song, W. & Yang, HL. Trees with 2-Reinforcement Number Three. Bull. Malays. Math. Sci. Soc. 39, 821–838 (2016). https://doi.org/10.1007/s40840-015-0140-2

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  • DOI: https://doi.org/10.1007/s40840-015-0140-2

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