Total and paired domination numbers of $$C_m$$ C m bundles over a cycle $$C_n$$ C n

Fu-Tao Hu, Moo Young Sohn () and Xue-gang Chen
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Fu-Tao Hu: Anhui University
Moo Young Sohn: Changwon National University
Xue-gang Chen: North China Electric Power University

Journal of Combinatorial Optimization, 2016, vol. 32, issue 2, No 16, 608-625

Abstract: Abstract Let $$G=(V,E)$$ G = ( V , E ) be a simple graph without isolated vertices. A set $$S$$ S of vertices is a total dominating set of a graph $$G$$ G if every vertex of $$G$$ G is adjacent to some vertex in $$S$$ S . A paired dominating set of $$G$$ 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 $$C_n\square C_m$$ C n □ C m for $$m=3,4$$ 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 $$C_m$$ C m bundles over a cycle $$C_n$$ C n for $$m=3,4$$ m = 3 , 4 . Moreover, we give the exact value for the total domination number of Cartesian product $$C_n\square C_5$$ C n □ C 5 and some upper bounds of $$C_m$$ C m bundles over a cycle $$C_n$$ C n where $$m\ge 5$$ m ≥ 5 .

Keywords: Cartesian product; Graph bundle; Total domination number; Paired domination number; 05C69 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10878-015-9885-7

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