On the Independent Double Roman Domination in Graphs

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 \(2a + 1 \le b \le 3a\).

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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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Correspondence to Doost Ali Mojdeh.

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Communicated by Hamid Reza Ebrahimi Vishki.

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Mojdeh, D.A., Mansouri, Z. On the Independent Double Roman Domination in Graphs. Bull. Iran. Math. Soc. 46, 905–915 (2020). https://doi.org/10.1007/s41980-019-00300-9

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Keywords

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

Mathematics Subject Classification

  • 05C69
  • 05C5