Skip to main content
Springer Nature Link
Log in

Total k-domination in Cartesian product graphs

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Let \(G=(V,E)\) be a graph. A set \(S\subseteq V\) is a total k-dominating set if every vertex \(v\in V\) has at least k neighbors in S. The total k-domination number \(\gamma _{kt}(G)\) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of the Cartesian product of two graphs, and we investigate the relationship between the total k-domination number of the Cartesian product graph with respect to the total k-domination number in the factors of the product. We also study the total k-domination number in certain particular cases of Cartesian products of graphs and determine the exact values of this parameter.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. S. Bermudo, D.L. Jalemskaya, J.M. Sigarreta, Total 2-domination in Grid graphs. Accepted (2016)

  2. S. Bermudo, J.C. Hernández-Gómez, J.M. Sigarreta, On the total k-domination in graphs. Accepted (2016)

  3. Y. Caro, R. Yuster, Dominating a family of graphs with small connected subgraphs. Comb. Probab. Comput. 9(4), 309–313 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. W.E. Clark, S. Suen, An inequality related to Vizing conjecture. Electron. J. Comb. 7(Note 4), 3pp (2000)

  5. E.J. Cockayne, R. Dawes, S.T. Hedetniemi, Total domination in graphs. Networks 10(3), 211–219 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Dorbec, S. Gravier, S. Klavžar, S. Špacapan, Some results on total domination in direct products of graphs. Discuss. Math. Graph Theory 26(1), 103–112 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. O. Favaron, M.A. Henning, J. Puech, D. Rautenbach, On domination and annihilation in graphs with claw-free blocks. Discrete Math. 231(1–3), 143–151 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Fernau, J.A. Rodríguez-Velázquez, J.M. Sigarreta, Global powerful \(r\)-alliances and total \(k\)-domination in graphs. Util. Math. 98, 127–147 (2015)

    MathSciNet  MATH  Google Scholar 

  9. S. Gravier, Total domination number of grid graphs. Discrete Appl. Math. 121(1–3), 119–128 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Harary, T.W. Haynes, Double domination in graphs. Ars Comb. 55, 201–213 (2000)

    MathSciNet  MATH  Google Scholar 

  11. M.A. Henning, A.P. Kazemi, \(k\)-Tuple total domination in graphs. Discrete Appl. Math. 158(9), 1006–1011 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. M.A. Henning, A.P. Kazemi, \(k\)-Tuple total domination number of cross products graphs. J. Comb. Optim. 4(3), 339–346 (2012)

    Article  MATH  Google Scholar 

  13. M.A. Henning, D.F. Rall, On the Total Domination Number of Cartesian Products of Graphs. Graphs Comb. 21(1), 63–69 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. M.A. Henning, A. Yeo, Total Domination in Graphs. Springer Monographs in Mathematics (Springer, Berlin, 2013)

  15. P.T. Ho, A note on the total domination number. Util. Math. 77, 97–100 (2008)

    MathSciNet  MATH  Google Scholar 

  16. X. Hou, Total domination of Cartesian products of graphs. Discuss. Math. Graph Theory 27(1), 175–178 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Klobucar, Total domination numbers of cartesian products. Math. Commun. 9(1), 35–44 (2004)

    MathSciNet  MATH  Google Scholar 

  18. V. Kulli, On n-total domination number in graphs, in Graph Theory, Combinatorics, Algorithms, and Applications (SIAM, Philadelphia, 1991), pp. 319–324

  19. N. Li, X. Hou, On the total \(\{k\}\)-domination number of Cartesian products of graphs. J. Comb. Optim. 18(2), 173–178 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. V.G. Vizing, Some unsolved problems in graph theory. Uspehi Mat Nauk 23(6(144)), 117–134 (1968)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This Research was partly supported by Plan Nacional I+D+I Grants MTM2015-70531-P and MTM 2013–46374-P (Spain), Junta de Andalucía FQM-260 (Spain) and CONACYT (FOMIX-CONACYT-UAGro-249818) (Mexico).

Author information

Authors and Affiliations

  1. Department of Economics, Quantitative Methods and Economic History, Universidad Pablo de Olavide, Carretera de Utrera Km. 1, 41013, Seville, Spain

    S. Bermudo

  2. Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, Mexico

    J. L. Sanchéz & J. M. Sigarreta

  3. Faculty of Computer Science and Mathematics, Holguín University, 80100, Holguín, Cuba

    J. L. Sanchéz

Authors
  1. S. Bermudo
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. J. L. Sanchéz
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. J. M. Sigarreta
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to S. Bermudo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bermudo, S., Sanchéz, J.L. & Sigarreta, J.M. Total k-domination in Cartesian product graphs. Period Math Hung 75, 255–267 (2017). https://doi.org/10.1007/s10998-017-0191-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-017-0191-2

Keywords

Mathematics Subject Classification