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Double Roman Domination in Digraphs

  • Guoliang Hao
  • Xiaodan ChenEmail author
  • Lutz Volkmann
Article
  • 140 Downloads

Abstract

Let D be a finite and simple digraph with vertex set V(D). A double Roman dominating function (DRDF) on a digraph D is a function \(f:V(D)\rightarrow \{0,1,2,3\}\) satisfying the condition that if \(f(v)=0\), then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3, while if \(f(v)=1\), then the vertex v must have at least one in-neighbor assigned 2 or 3. The weight of a DRDF f is the sum \(\sum _{v\in V(D)}f(v)\). The double Roman domination number of a digraph D is the minimum weight of a DRDF on D. In this paper, we initiate the study of the double Roman domination of digraphs, and we give several relations between the double Roman domination number of a digraph and other domination parameters such as Roman domination number, k-domination number and signed domination number. Moreover, various bounds on the double Roman domination number of a digraph are presented, and a Nordhaus–Gaddum type inequality for the parameter is also given.

Keywords

Double Roman domination Roman domination k-domination Signed domination Digraph Nordhaus–Gaddum 

Mathematics Subject Classification

05C20 05C69 

Notes

Acknowledgements

The first author was supported by the Research Foundation of Education Bureau of Jiangxi Province of China (No. GJJ150561) and the Doctor Fund of East China University of Technology (No. DHBK2015319). The second author was supported partially by National Natural Science Foundation of China (No. 11501133) and Guangxi Natural Science Foundation (Nos. 2014GXNSFBA118008, 2016GXNSFAA380293).

References

  1. 1.
    Ahangar, H.A., Henning, M.A., Löwenstein, C., Zhao, Y., Samodivkin, V.: Signed Roman domination in graphs. J. Comb. Optim. 27, 241–255 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amjadi, J., Falahat, M., Sheikholeslami, S.M., Rad, N.J.: Strong equality between the 2-rainbow domination and independent 2-rainbow domination numbers in trees. Bull. Malays. Math. Sci. Soc. 39, 205–218 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beeler, R.A., Haynes, T.W., Hedetniemi, S.T.: Double Roman domination. Discrete Appl. Math. 211, 23–29 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caro, Y., Henning, M.A.: Directed domination in oriented graphs. Discrete Appl. Math. 160, 1053–1063 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chambers, E.W., Kinnersley, B., Prince, N., West, D.B.: Extremal problems for Roman domination. SIAM J. Discrete Math. 23, 1575–1586 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chellali, M., Haynes, T.W., Hedetniemi, S.T.: Lower bounds on the Roman and independent Roman domination numbers. Appl. Anal. Discrete Math. 10, 65–72 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fu, Y.: Dominating set and converse dominating set of a directed graph. Am. Math. Mon. 75, 861–863 (1968)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gyaürki, Š.: On the difference of the domination number of a digraph and of its reverse. Discrete Appl. Math. 160, 1270–1276 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hao, G., Qian, J.: Bounds on the domination number of a digraph. J. Comb. Optim. (2017).  https://doi.org/10.1007/s10878-017-0154-9 zbMATHGoogle Scholar
  10. 10.
    Harary, F., Norman, R.Z., Cartwright, D.: Structural Models. Wiley, New York (1965)zbMATHGoogle Scholar
  11. 11.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, Inc., New York (1998)zbMATHGoogle Scholar
  12. 12.
    Karami, H., Sheikholeslami, S.M., Khodkar, A.: A Lower bounds on the signed domination numbers of directed graphs. Discrete Math. 309, 2567–2570 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Shao, Z., Li, Z., Peperko, A., Wan, J., Žerovnik, J.: Independent rainbow domination of graphs. Malays. Math. Sci. Soc, Bull (2017).  https://doi.org/10.1007/s40840-017-0488-6 zbMATHGoogle Scholar
  14. 14.
    Sheikholeslami, S.M., Volkmann, L.: The Roman domination number of a digraph. Acta Univ. Apulensis 27, 77–86 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Volkmann, L.: Signed domination and signed domatic numbers of digraphs. Discuss. Math. Graph Theory 31, 415–427 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zelinka, B.: Signed domination numbers of directed graphs. Czech. Math. J. 55, 479–482 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.College of ScienceEast China University of TechnologyNanchangPeople’s Republic of China
  2. 2.College of Mathematics and Information ScienceGuangxi UniversityNanningPeople’s Republic of China
  3. 3.Lehrstuhl II für MathematikRWTH Aachen UniversityAachenGermany

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