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Mixed Roman Domination in Graphs

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Abstract

Let \(G = (V, E)\) be a simple graph with vertex set V and edge set E. A mixed Roman dominating function (MRDF) of G is a function \(f: V\cup E\rightarrow \{0,1,2\}\) satisfying the condition every element \(x\in V\cup E\) for which \(f(x)= 0\) is adjacent or incident to at least one element \(y\in V\cup E\) for which \(f(y) = 2\). The weight of a MRDF f is \(\omega (f)=\sum _{x\in V\cup E}f(x)\). The mixed Roman domination number of G is the minimum weight of a mixed Roman dominating function of G. In this paper, we initiate the study of the mixed Roman domination number and we present bounds for this parameter. We characterize the graphs attaining an upper bound and the graphs having small mixed Roman domination numbers.

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References

  1. Adabi, M., Ebrahimi Targhi, E., Jafari Rad, N., Moradi, M.S.: Properties of independent Roman domination in graphs. Australas. J. Combin. 52, 11–18 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Alavi, Y., Behzad, M., Lesniak, L., Nordhaus, E.A.: Total matchings and total coverings of graphs. J. Graph Theory 1, 135–140 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alavi, Y., Liu, J.Q., Wang, J.F., Zhang, Z.F.: On total covers of graphs. Discret. Math. 100(13), 229–233 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bermudo, S., Fernau, H., Sigarreta, J.M.: The differential and the Roman domination number of a graph. Appl. Anal. Discret. Math. 8, 155–171 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chellali, M., Jafari Rad, N.: A note on the independent Roman domination in unicyclic graphs. Opusc. Math. 32, 715–718 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chellali, M., Jafari Rad, N.: Strong equality between the Roman domination and independent Roman domination numbers in trees. Discuss. Math. Graph Theory 33, 337–346 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cockayne, E.J., Goodman, S., Hedetniemi, S.: A linear algorithm for the domination number of a tree. Inf. Process. Lett. 4, 41–44 (1975)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Ebrahimi Targhi, E., Jafari Rad, N., Mynhardt, C.M., Wu, Y.: Bounds for the independent Roman domination number in graphs. J. Combin. Math. Combin. Comput. 80, 351–365 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Erdös, P., Meir, A.: On total matching numbers and total covering numbers of complementary graphs. Discret. Math. 19, 229–233 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hatami, P.: An approximation algorithm for the total covering problem. Discuss. Math. Graph Theory 27(3), 553–558 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  13. Hedetniemi, S.T., Rubalcaba, R.R., Slater, P.J., Walsh, M.: Few compare to the great Roman empire. Congr. Numer. 217, 129–136 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Lan, J.K., Chang, G.J.: On the mixed domination problem in graphs. Theor. Comput. Sci. 476, 84–93 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  17. West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)

    Google Scholar 

  18. Zhao, Y., Kang, L., Sohn, M.Y.: The algorithmic complexity of mixed domination in graphs. Theor. Comput. Sci. 412, 2387–2392 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors thanks the referees for their helpful comments and suggestions which helped improve the exposition and readability of the paper. Research of the first author was supported in part by the Babol University of Technology. Research of the second author was supported in part by the University of Johannesburg. Research of the third author was supported by the Ministry of Education and Science MTM2011-28800-C02-02, Spain.

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Authors and Affiliations

  1. Department of Basic Science, Babol University of Technology, Babol, Iran

    H. Abdollahzadeh Ahangar

  2. Department of Mathematics, East Tennessee State University, Johnson City, TN, 37614, USA

    Teresa W. Haynes

  3. Department of Mathematics, University of Johannesburg, Auckland Park, South Africa

    Teresa W. Haynes

  4. Department of Mathematics, University of Cadiz, Cádiz, Spain

    J. C. Valenzuela-Tripodoro

Authors
  1. H. Abdollahzadeh Ahangar
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  2. Teresa W. Haynes
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  3. J. C. Valenzuela-Tripodoro
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Correspondence to H. Abdollahzadeh Ahangar.

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Communicated by Xueliang Li.

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Ahangar, H.A., Haynes, T.W. & Valenzuela-Tripodoro, J.C. Mixed Roman Domination in Graphs. Bull. Malays. Math. Sci. Soc. 40, 1443–1454 (2017). https://doi.org/10.1007/s40840-015-0141-1

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  • DOI: https://doi.org/10.1007/s40840-015-0141-1

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