Mixed Roman Domination in Graphs

  • H. Abdollahzadeh Ahangar
  • Teresa W. Haynes
  • J. C. Valenzuela-Tripodoro
Article

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.

Keywords

Roman dominating function Roman domination number   Mixed Roman dominating function Mixed Roman domination number 

Mathematics Subject Classification

05C69 

Notes

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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Copyright information

© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Authors and Affiliations

  • H. Abdollahzadeh Ahangar
    • 1
  • Teresa W. Haynes
    • 2
    • 3
  • J. C. Valenzuela-Tripodoro
    • 4
  1. 1.Department of Basic ScienceBabol University of TechnologyBabolIran
  2. 2.Department of MathematicsEast Tennessee State UniversityJohnson CityUSA
  3. 3.Department of MathematicsUniversity of JohannesburgAuckland ParkSouth Africa
  4. 4.Department of MathematicsUniversity of CadizCádizSpain

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