Abstract
Let \(G=(V,E)\) be a graph. A subset \(S\subseteq V\) is a k-dominating set of G if each vertex in \(V-S\) is adjacent to at least k vertices in S. The k-domination number of G is the cardinality of the smallest k-dominating set of G. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs \(P(5k+1, 2)\) and \(P(5k+2, 2)\), for \(k>0\), is \(4k+2\) and \(4k+3\), respectively. This proves two conjectures due to Cheng (Ph.D. thesis, National Chiao Tung University, 2013). Moreover, we determine the exact 2-domination number of generalized Petersen graphs P(2k, k) and \(P(5k+4,3)\). Furthermore, we give a good lower and upper bounds on the 2-domination number of generalized Petersen graphs \(P(5k+1, 3), P(5k+2,3)\) and \(P(5k+3, 3).\)
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The authors would like to thank the reviewers for their comments to improve the paper and clarify some of the proofs.
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Communicating Editor: Sharad S Sane
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Bakhshesh, D., Farshi, M. & Hooshmandasl, M.R. 2-Domination number of generalized Petersen graphs. Proc Math Sci 128, 17 (2018). https://doi.org/10.1007/s12044-018-0395-2
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DOI: https://doi.org/10.1007/s12044-018-0395-2