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Edge-Removal and Edge-Addition in \(\alpha \)-Domination

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Abstract

Let \(G=(V,E)\) be any graph without isolated vertices. For some \(\alpha \) with \(0<\alpha \le 1\) and a dominating set S of G, we say that S is an \(\alpha \)-dominating set if for any \(v\in V-S, |N(v)\cap S| \ge \alpha |N(v)|\). The cardinality of a smallest \(\alpha \)-dominating set of G is called the \(\alpha \) -domination number of G and is denoted by \(\gamma _{\alpha }(G)\). In this paper, we study graphs G for which the removal of any edge or the addition of any further edge affects the \(\alpha \)-domination number of G.

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Acknowledgments

We would like to thank the referees for their careful review of the paper and helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Shahrood University, Shahrood, Iran

    Nader Jafari Rad

  2. Lehrstuhl II für Mathematik, RWTH Aachen University, 52056, Aachen, Germany

    Lutz Volkmann

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  1. Nader Jafari Rad
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  2. Lutz Volkmann
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Correspondence to Nader Jafari Rad.

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Rad, N.J., Volkmann, L. Edge-Removal and Edge-Addition in \(\alpha \)-Domination. Graphs and Combinatorics 32, 1155–1166 (2016). https://doi.org/10.1007/s00373-015-1614-6

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  • DOI: https://doi.org/10.1007/s00373-015-1614-6

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