Advertisement

Extremal Digraphs for an Upper Bound on the Double Roman Domination Number

  • Lyes OuldrabahEmail author
  • Mostafa Blidia
  • Ahmed Bouchou
  • Lutz Volkmann
Article
  • 24 Downloads

Abstract

A doubleRoman dominating function of a digraph D is a function \(f:V\longrightarrow \{0,1,2,3\}\) such that every vertex u for which \(f(u)=0\) has an in-neighbor v for which \(f(v)=3\) or at least two in-neighbors assigned 2 under f, while if \(f(u)=1\), then the vertex u must have at least one in-neighbor assigned 2 or 3. The weight of a double Roman dominating function is the value \(f(V)=\sum _{u\in V}f(u)\). The minimum weight of a double Roman dominating function of a digraph D is called the double Roman domination number of D, denoted by \(\gamma _{dR}\left( D\right) \). This paper gives a descriptive characterization for some classes of digraphs satisfying \(\gamma _{dR}(D)=2\left( n-\Delta ^{+}(D)\right) +1\). Also, a descriptive characterization for digraphs D of order \(n\ge 4\) for which \(\gamma _{dR}(D)+\gamma _{dR}({\overline{D}})=2n+3\) holds where \({\overline{D}}\) is the complement of D.

Keywords

Double Roman domination Digraph Oriented Graph Tournament 

Mathematics Subject Classification

05C20 05C69 

Notes

Acknowledgements

We would like to thank the referees for many useful suggestions that have considerably improved on the presentation of the paper.

References

  1. 1.
    Ahangar, H.A., Chellali, M., Sheikholeslami, S.M.: On the double Roman domination in graphs. Discrete Appl. Math. 232, 1–7 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amjadi, J., Nazari-Moghaddam, S., Sheikholeslami, S.M., Volkmann, L.: An upper bound on the double Roman domination number. J. Comb. Optim. 36, 81–89 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anu, V., Lakshmanan, S.A.: Double Roman domination number. Discrete Appl. Math. 244, 198–204 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beeler, R.A., Haynes, T.W., Hedetniemi, S.T.: Double Roman domination. Discrete Appl. Math. 211, 23–29 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hao, G., Chen, X., Volkmann, L.: Double Roman domination of digraphs. Accepted for publication in Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia (2017)Google Scholar
  6. 6.
    Volkmann, L.: Double Roman domination and domatic numbers of graphs. Commun. Comb. Optim. 3, 71–77 (2018)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Yue, J., Wei, M., Li, M., Liu, G.: On the double Roman domination of graphs. Appl. Math. Comput. 338, 669–675 (2018)MathSciNetGoogle Scholar
  8. 8.
    Zhang, X., Li, Z., Jiang, H., Shao, Z.: Double Roman domination in trees. Inf. Process. Lett. 134, 31–34 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  • Lyes Ouldrabah
    • 1
    Email author
  • Mostafa Blidia
    • 1
  • Ahmed Bouchou
    • 2
  • Lutz Volkmann
    • 3
  1. 1.Department of MathematicsUniversity of Blida 1BlidaAlgeria
  2. 2.Department of MathematicsUniversity of MédéaMedeaAlgeria
  3. 3.Lehrstuhl II für MathematikRWTH Aachen UniversityAachenGermany

Personalised recommendations