Abstract
Let \(G=(V,E)\) be a simple graph without isolated vertices. A set \(S\) of vertices is a total dominating set of a graph \(G\) if every vertex of \(G\) is adjacent to some vertex in \(S\). A paired dominating set of \(G\) is a dominating set whose induced subgraph has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired dominating set) is the total domination number (respectively, the paired domination number). Hu and Xu (J Combin Optim 27(2):369–378, 2014) computed the exact values of total and paired domination numbers of Cartesian product \(C_n\square C_m\) for \(m=3,4\). Graph bundles generalize the notions of covering graphs and Cartesian products. In this paper, we generalize these results given in Hu and Xu (J Combin Optim 27(2):369–378, 2014) to graph bundle and compute the total domination number and the paired domination number of \(C_m\) bundles over a cycle \(C_n\) for \(m=3,4\). Moreover, we give the exact value for the total domination number of Cartesian product \(C_n\square C_5\) and some upper bounds of \(C_m\) bundles over a cycle \(C_n\) where \(m\ge 5\).
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Acknowledgments
The authors would like to express their gratitude to the anonymous referees for their critical commons and helpful suggestions on the original manuscript, which resulted in this version. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2005115). The first author was supported by NNSF of China (No. 11401004), Anhui Provincial Natural Science Foundation (No. 1408085QA03), and the doctoral scientific research startup fund of Anhui University.
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Hu, FT., Sohn, M.Y. & Chen, Xg. Total and paired domination numbers of \(C_m\) bundles over a cycle \(C_n\) . J Comb Optim 32, 608–625 (2016). https://doi.org/10.1007/s10878-015-9885-7
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DOI: https://doi.org/10.1007/s10878-015-9885-7