Some Progress on the Restrained Roman Domination

Abstract

A Roman dominating function on a graph G is a function \(f:V(G)\rightarrow \{0,1,2\}\) satisfying the condition that every vertex u for which \(f(u) = 0\) is adjacent to at least one vertex v for which \(f(v) =2\). A Roman dominating function f is called a restrained Roman dominating function if the induced subgraph of G by the vertices with label 0 has no isolated vertex. The weight of a restrained Roman dominating function is the value \(\omega (f)=\sum _{u\in V(G)} f(u)\). The minimum weight of a restrained Roman dominating function of G is called the restrained Roman domination number of G and denoted by \(\gamma _{rR}(G)\). In this paper, we show that for any graph G of order \(n\ge 5\), \(6\le \gamma _{rR} (G)+\gamma _{rR} ({\overline{G}})\le n+5\) and characterize all the extremal graphs. In addition, we classify all graphs with large restrained Roman domination number.

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References

  1. 1.

    Abdollahzadeh Ahangar, H., Amjadi, J., Atapour, M., Chellali, M., Sheikholeslami, S.M.: Double Roman trees. Ars Comb. 145, 173–183 (2019)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Abdollahzadeh Ahangar, H., Álvarez, M.P., Chellali, M., Sheikholeslami, S.M., Valenzuela-Tripodoro, J.C.: Triple Roman domination in graphs. Appl. Math. Comput. 386C, 125444 (2020)

  3. 3.

    Abdollahzadeh Ahangar, H., Amjadi, J., Chellali, M., Nazari-Moghaddam, S., Sheikholeslami, S.M.: Trees with double Roman domination number twice the domination number plus two. Iran. J. Sci. Technol. Trans. A Sci 43, 1081–1088 (2019)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Abdollahzadeh Ahangar, H., Chellali, M., Sheikholeslami, S.M.: On the double Roman domination in graphs. Discrete Appl. Math. 103, 245–258 (2017)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Abdollahzadeh Ahangar, H., Chellali, M., Sheikholeslami, S.M.: Outer independent double Roman domination. Appl. Math. Comput. 364, 124617 (2020). (9 pages)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Abdollahzadeh Ahangar, H., Chellali, M., Sheikholeslami, S.M., Valenzuela-Tripodoro, J.C.: Total Roman \(\{2\}\)-domination in graphs. Discuss. Math. Graph Theory, to appear

  7. 7.

    Abdollahzadeh Ahangar Teresa, H., Haynes, W., Tripodoro, Juan Carlos Valenzuela: Mixed Roman domination in graphs. Bull. Malays. Math. Sci. Soc. 40, 1443–1454 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Abdollahzadeh Ahangar, H., Khoeilar, R., Shabazi, L., Sheikholeslami, S.M.: Bounds on Signed total double Roman domination. Commun. Comb. Optim. 5, 191–206 (2020)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Abdollahzadeh Ahangar, H., Mirmehdipour, S.R.: Bounds on the restrained Roman domination number of a graph. Commun. Comb. Optim. 1, 75–82 (2016)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Chellali, M., Jafari Rad, N., Sheikholeslami, S.M., Volkmann, L.: Roman domination in graphs. In: Haynes, T.W., Hedetniemi, S.T., Henning, M.A., (Eds), Topics in domination in graphs, Springer, Switzerland (2020)

  11. 11.

    Chellali, M., Jafari Rad, N., Sheikholeslami, S.M., Volkmann, L.: Varieties of Roman domination. In: Haynes, T.W., Hedetniemi, S.T., Henning, M.A., (Eds) structures of domination in graphs, Springer, Switzerland (2020)

  12. 12.

    Chellali, M., Jafari Rad, N., Sheikholeslami, S.M., Volkmann, L.: Varieties of Roman domination II, AKCE Int. J. Graphs Comb., in press

  13. 13.

    Chellai, M., Jafari Rad, N., Sheikholeslami, S.M., Volkmann, L.: The Roman domatic problem in graphs and digraphs: A survey, Discuss. Math. Graph Theory, in press

  14. 14.

    Chellai, M., Jafari Rad, N., Sheikholeslami, S.M., Volkmann, L.: A survey on Roman domination parameters in directed graphs, J. Combin. Math. Combin. Comput. (to appear)

  15. 15.

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

    MathSciNet  Article  Google Scholar 

  16. 16.

    Domke, G.S., Hattingh, J.H., Hedetniemi, S.T., Markus, L.R.: Restrained domination in trees. Discrete Math. 211, 1–9 (2000)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Domke, G.S., Hattingh, J.H., Henning, M.A., Markus, L.R.: Restrained domination in graphs with minimum degree two. J. Comb. Math. Comb. Comput. 35, 239–254 (2000)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Domke, G.S., Hattingh, J.H., Hedetniemi, S.T., Laskar, R.C., Markus, L.R.: Restrained domination in graphs. Discrete Math. 203, 61–69 (1999)

    MathSciNet  Article  Google Scholar 

  19. 19.

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

    Google Scholar 

  20. 20.

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

    Google Scholar 

  21. 21.

    Nordhaus, E.A., Gaddum, J.W.: On complementary graphs. Am. Math. Mon. 63, 175–177 (1956)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Pushpam, P.R.L., Padmapriea, S.: Restrained Roman domination in graphs. Trans. Comb. 4, 1–17 (2015)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Jafari Rad, N., Krzywkowski, M.: On the restrained Roman domination in graphs, Manuscript

  24. 24.

    Revelle, C.S., Rosing, K.E.: Defendens imperium romanum: a classical problem in military strategy. Am. Math. Mon. 107(7), 585–594 (2000)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Shi, Y., Wei, M., Yue, J., Zhao, Y.: Coupon coloring of some special graphs. J. Comb. Optim. 33, 156–164 (2017)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Stewart, I.: Defend the Roman Empire. Sci. Am. 281(6), 136–139 (1999)

    Article  Google Scholar 

  27. 27.

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

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Yue, J., Wei, M., Li, M., Liu, G.: On the double Roman domination of graphs. Appl. Math. Comput. 338, 669–675 (2018)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the referees for their helpful comments and suggestions to improve the exposition and readability of the paper. The second author was supported by the Babol Noshirvani University of Technology under research grant number BNUT/385001/99.

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Correspondence to H. Abdollahzadeh Ahangar.

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Siahpour, F., Abdollahzadeh Ahangar, H. & Sheikholeslami, S.M. Some Progress on the Restrained Roman Domination. Bull. Malays. Math. Sci. Soc. 44, 733–751 (2021). https://doi.org/10.1007/s40840-020-00965-0

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Keywords

  • Roman domination number
  • Restrained Roman domination number
  • Nordhaus–Gaddum inequalities

Mathematics Subject Classification

  • 05C69
  • 05C05