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

Author

Listed:
  • Fu-Tao Hu

    (Anhui University)

  • Moo Young Sohn

    (Changwon National University)

  • Xue-gang Chen

    (North China Electric Power University)

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 .

Suggested Citation

  • Fu-Tao Hu & Moo Young Sohn & Xue-gang Chen, 2016. "Total and paired domination numbers of $$C_m$$ C m bundles over a cycle $$C_n$$ C n," Journal of Combinatorial Optimization, Springer, vol. 32(2), pages 608-625, August.
    • Handle: RePEc:spr:jcomop:v:32:y:2016:i:2:d:10.1007_s10878-015-9885-7
    DOI: 10.1007/s10878-015-9885-7
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    References listed on IDEAS

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    1. Juan Liu & Xindong Zhang & Jixiang Meng, 2011. "On domination number of Cartesian product of directed paths," Journal of Combinatorial Optimization, Springer, vol. 22(4), pages 651-662, November.
    2. Fu-Tao Hu & Jun-Ming Xu, 2014. "Total and paired domination numbers of toroidal meshes," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 369-378, February.
    3. Michel Mollard, 2014. "The domination number of Cartesian product of two directed paths," Journal of Combinatorial Optimization, Springer, vol. 27(1), pages 144-151, January.
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