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Total and Paired Domination Numbers of Cylinders

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Let \(\gamma _{t}(P_p\square C_q)\) and \(\gamma _{pr}(P_p\square C_q)\) be the total domination number and the paired domination number, respectively, of cylinders, that is, the Cartesian product of a path \(P_p\) and a cycle \(C_q\) for \(p\ge 2\) and \(q\ge 3\). Hu et al. (J Comb Optim 32(2):608–625, 2016) determined the exact values of \(\gamma _{t}(P_p\square C_q)\) and \(\gamma _{pr}(P_p\square C_q)\) for \(p\ge 2\) and \(q\in \{3,4\}\). In this paper, we compute their exact values for \(p\in \{2,3,4\}\) and \(q\ge 5\) and obtain the upper and the lower bounds for other values of p and q.

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References

  1. Carreño, J.J., Martínez, J.A., Puertas, M.L.: A general lower bound for the domination number of cylindrical graphs. Bull. Malays. Math. Sci. Soc. 43(2), 1671–1684 (2020). https://doi.org/10.1007/s40840-019-00765-1

    Article  MathSciNet  MATH  Google Scholar 

  2. Chang, T.Y., Clark, W.E.: The domination numbers of the \(5\times n\) and \(6\times n\) grid graphs. J. Graph Theory 17, 81–108 (1993). https://doi.org/10.1002/jgt.3190170110

    Article  MathSciNet  MATH  Google Scholar 

  3. Cockayne, E.J., Dawes, R.M., Hedetniemi, S.T.: Total domination in graphs. Networks 10(3), 211–219 (1980). https://doi.org/10.1002/net.3230100304

    Article  MathSciNet  MATH  Google Scholar 

  4. Gavlas, H., Schultz, K.: Efficient open domination. Electron. Notes Discrete Math. 11, 681–691 (2002). https://doi.org/10.1016/S1571-0653(04)00113-1

    Article  MathSciNet  MATH  Google Scholar 

  5. Gravier, S.: Total domination number of grid graphs. Discrete Appl. Math. 121, 119–128 (2002). https://doi.org/10.1016/S0166-218X(01)00297-9

    Article  MathSciNet  MATH  Google Scholar 

  6. Gonçalves, D., Pinlou, A., Rao, M., Thomassé, S.: The domination number of grids. SIAM J. Discrete Math. 25(3), 1443–1453 (2011). https://doi.org/10.1137/11082574

    Article  MathSciNet  MATH  Google Scholar 

  7. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)

    MATH  Google Scholar 

  8. Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.): Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998)

  9. Haynes, T.W., Slater, P.J.: Paired-domination in graphs. Networks 32, 199–206 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hu, F.T., Sohn, M.Y., Chen, X.G.: Total and paired domination numbers of \(C_m\) bundles over a cycle \(C_n\). J. Comb. Optim. 32(2), 608–625 (2016). https://doi.org/10.1007/s10878-015-9885-7

    Article  MathSciNet  MATH  Google Scholar 

  11. Hu, F.T., Xu, J.M.: Total and paired domination numbers of toroidal meshes. J. Comb. Optim. 27(2), 369–378 (2014). https://doi.org/10.1007/s10878-012-9519-2

    Article  MathSciNet  MATH  Google Scholar 

  12. Jacobson, M.S., Kinch, L.F.: On the domination number of products of graphs: I. Ars Combin. 18, 33–44 (1983)

    MathSciNet  MATH  Google Scholar 

  13. Klavžar, S., Seifter, N.: Dominating Cartesian products of cycles. Discrete Appl. Math. 59(2), 129–136 (1995). https://doi.org/10.1016/0166-218X(93)E0167-W

    Article  MathSciNet  MATH  Google Scholar 

  14. Klobučar, A.: Total domination number of Cartesian products. Math. Commun. 9(1), 35–44 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Kuziak, D., Peterin, I., Yero, I.G.: Erratum to “Total domination number of Cartesian products" [Math. Commun. 9(2004), 35–44]. Math. Commun. 19(1), 195–200 (2014)

  16. Nandi, M., Parui, S., Adhikari, A.: The domination numbers of cylindrical grid graphs. Appl. Math. Comput. 217(10), 4879–4889 (2011). https://doi.org/10.1016/j.amc.2010.11.019

    Article  MathSciNet  MATH  Google Scholar 

  17. Pavlič, P., Žerovnik, J.: A note on the domination number of the Cartesian products of paths and cycles. Kragujevac J. Math. 37(2), 275–285 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Proffitt, K.E., Haynes, T.W., Slater, P.J.: Paired-domination in grid graphs. Congr. Numer. 150, 161–172 (2001)

    MathSciNet  MATH  Google Scholar 

  19. West, D.B.: Introduction to Graph Theory, 2nd edn. Pearson, London (2001)

    Google Scholar 

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Acknowledgements

I would like to thank Canada-ASEAN Scholarships and Educational Exchanges for Development (SEED) 2021-2022 provided with the support of Global Affairs Canada. This provided an excellent opportunity to me for doing research at the University of Victoria, Canada. I also wish to thank Kieka Mynhardt for her helpful comments and suggestions.

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  1. Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani, 12120, Thailand

    Pannawat Eakawinrujee

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  1. Pannawat Eakawinrujee
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Communicated by Ismael G. Yero.

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Eakawinrujee, P. Total and Paired Domination Numbers of Cylinders. Bull. Malays. Math. Sci. Soc. 45, 3321–3334 (2022). https://doi.org/10.1007/s40840-022-01382-1

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