Skip to main content
SpringerLink
Log in

Roman Domination and Double Roman Domination Numbers of Sierpiński Graphs \(S(K_n,t)\)

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

Different types of domination on the Sierpiński graphs \(S(K_n,t)\) will be studied in this paper. More precisely, we propose a minimal dominating set for each \(S(K_n,t)\) so that the exact values of its Roman domination numbers and double Roman domination numbers are given. As applications, some previous bounds and results are confirmed to be tight and further generalized.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Ahangar, H.A., Henning, M.A., Samodivkin, V., Yero, I.G.: Total Roman domination in graphs. Appl. Anal. Discrete Math. 10, 501–517 (2016)

    Article  MathSciNet  Google Scholar 

  2. Anu, V., Aparna Lakshmanan, S.: Double Roman domination number. Discret. Appl. Math. 244, 198–204 (2018)

    Article  MathSciNet  Google Scholar 

  3. Anu, V., Aparna Lakshmanan, S.: The double Roman domination number of generalized Sierpiński graphs. Discret. Math. Algorith. Appl. 2020, 2050047 (2020)

    Article  Google Scholar 

  4. Beeler, R.A., Haynes, T.W., Hedetniemi, S.T.: Double Roman domination. Discret. Appl. Math. 211, 23–29 (2016)

    Article  MathSciNet  Google Scholar 

  5. Chambers, E.W., Kinnersley, B., Prince, N., West, D.B.: Extremal problems for Roman domination. SIAM J. Discret. Math. 23, 1575–1586 (2009)

    Article  MathSciNet  Google Scholar 

  6. Cockayne, E.J., Dreyer, P.A., Hedetniemi, S.M., Hedetniemi, S.T.: On Roman domination in graphs. Discret. Math. 278, 11–22 (2004)

    Article  MathSciNet  Google Scholar 

  7. Gary, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman & Co., New York (1979)

    Google Scholar 

  8. Gravier, S., Klavžar, S., Mollard, M.: Codes and \(L(2,1)\)-labelings in Sierpiński graphs. Taiwan. J. Math. 9, 671–681 (2005)

    Article  Google Scholar 

  9. Hinz, A.M., auf der Heide, C.: An efficient algorithm to determine all shortest paths in Sierpiński graphs. Discret. Appl. Math. 177, 111–120 (2014)

    Article  Google Scholar 

  10. Hinz, A.M., Klavžar, S., Zemljič, S.S.: A survey and classification of Sierpiński-type graphs. Discret. Appl. Math. 217, 565–600 (2017)

    Article  Google Scholar 

  11. Hinz, A.M., Klavžar, S., Petr, C.: The Tower of Hanoi—Myths and Maths, 2nd edn. Birkhäuser, Cham (2018)

    MATH  Google Scholar 

  12. Klavžar, S., Milutinović, U.: Graphs \(S(n, k)\) and a variant of the Tower of Hanoi problem. Czechoslov. Math. J. 47, 95–104 (1997)

    Article  MathSciNet  Google Scholar 

  13. Klavžar, S., Milutinović, U., Petr, C.: 1-perfect codes in Sierpiński graphs. Bull. Aust. Math. Soc. 66, 369–384 (2002)

    Article  Google Scholar 

  14. Klavžar, S., Peterin, I., Zemljič, S.S.: Hamming dimension of a graph: the case of Sierpiński graphs. Eur. J. Comb. 34, 460–473 (2013)

    Article  Google Scholar 

  15. Ramezani, F., Rodríguez-Bazan, E.D., Rodríguez-Velázquez, J.A.: On the Roman domination number of generalized Sierpiński graphs. Filomat 31, 6515–6528 (2017)

    Article  MathSciNet  Google Scholar 

  16. Revelle, C.S., Rosing, K.E.: Defendens imperium romanum: a classical problem in military strategy. Am. Math. Mon. 107, 585–594 (2000)

    Article  MathSciNet  Google Scholar 

  17. Stewart, I.: Defend the Roman empire. Sci. Am. 281, 136–138 (1999)

    Article  Google Scholar 

  18. Volkmann, L.: Double Roman domination and domatic numbers of graphs. Commun. Comb. Optim. 3, 71–77 (2018)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We sincerely thank the anonymous reviewers for their insightful comments. This research is supported by the Ministry of Science and Technology of Taiwan R.O.C. under the project MOST 109-2115-M-031-006-MY2.

Author information

Authors and Affiliations

  1. Department of Mathematics, Soochow University, Taipei, 111002, Taiwan R.O.C.

    Chia-An Liu

Authors
  1. Chia-An Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chia-An Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Sandi Klavžar.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, CA. Roman Domination and Double Roman Domination Numbers of Sierpiński Graphs \(S(K_n,t)\). Bull. Malays. Math. Sci. Soc. 44, 4043–4058 (2021). https://doi.org/10.1007/s40840-021-01136-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-021-01136-5

Keywords

Mathematics Subject Classification

Search

Navigation