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On the Independent Double Roman Domination in Graphs

  • Doost Ali MojdehEmail author
  • Zhila Mansouri
Original Paper
  • 4 Downloads

Abstract

An independent double Roman dominating function (IDRDF) on a graph \(G=(V,E)\) is a function \(f{:}V(G)\rightarrow \{0,1,2,3\}\) having the property that if \(f(v)=0\), then the vertex v has at least two neighbors assigned 2 under f or one neighbor w assigned 3 under f, and if \(f(v)=1\), then there exists \(w\in N(v)\) with \(f(w)\ge 2\), such that the set of vertices with positive weight is independent. The weight of an IDRDF is the value \(\sum _{u\in V}f(u)\). The independent double Roman domination number \(i_\mathrm{dR}(G)\) of a graph G is the minimum weight of an IDRDF on G. We continue the study of the independent double Roman domination and show its relationships to both independent domination number (IDN) and independent Roman \(\{2\}\)-domination number (IR2DN). We present several sharp bounds on the IDRDN of a graph G in terms of the order of G, maximum degree and the minimum size of edge cover. Finally, we show that, any ordered pair (ab) is realizable as the IDN and IDRDN of some non-trivial tree if and only if \(2a + 1 \le b \le 3a\).

Keywords

Independent double Roman domination Independent Roman {2}-domination Independent domination Graphs 

Mathematics Subject Classification

05C69 05C5 

Notes

Acknowledgements

The authors sincerely thank the referees for their careful review of this paper and some useful comments and valuable suggestions.

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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MazandaranBabolsarIran

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