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An upper bound on the double Roman domination number

Author

Listed:
  • J. Amjadi

    (Azarbaijan Shahid Madani University)

  • S. Nazari-Moghaddam

    (Azarbaijan Shahid Madani University)

  • S. M. Sheikholeslami

    (Azarbaijan Shahid Madani University)

  • L. Volkmann

    (RWTH Aachen University)

Abstract

A double Roman dominating function (DRDF) on a graph $$G=(V,E)$$ G = ( V , E ) is a function $$f : V \rightarrow \{0, 1, 2, 3\}$$ f : V → { 0 , 1 , 2 , 3 } having the property that if $$f(v) = 0$$ f ( v ) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with $$f(w)=3$$ f ( w ) = 3 , and if $$f(v)=1$$ f ( v ) = 1 , then vertex v must have at least one neighbor w with $$f(w)\ge 2$$ f ( w ) ≥ 2 . The weight of a DRDF f is the value $$f(V) = \sum _{u \in V}f(u)$$ f ( V ) = ∑ u ∈ V f ( u ) . The double Roman domination number $$\gamma _{dR}(G)$$ γ dR ( G ) of a graph G is the minimum weight of a DRDF on G. Beeler et al. (Discrete Appl Math 211:23–29, 2016) observed that every connected graph G having minimum degree at least two satisfies the inequality $$\gamma _{dR}(G)\le \frac{6n}{5}$$ γ dR ( G ) ≤ 6 n 5 and posed the question whether this bound can be improved. In this paper, we settle the question and prove that for any connected graph G of order n with minimum degree at least two, $$\gamma _{dR}(G)\le \frac{8n}{7}$$ γ dR ( G ) ≤ 8 n 7 .

Suggested Citation

  • J. Amjadi & S. Nazari-Moghaddam & S. M. Sheikholeslami & L. Volkmann, 2018. "An upper bound on the double Roman domination number," Journal of Combinatorial Optimization, Springer, vol. 36(1), pages 81-89, July.
  • Handle: RePEc:spr:jcomop:v:36:y:2018:i:1:d:10.1007_s10878-018-0286-6
    DOI: 10.1007/s10878-018-0286-6
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    Cited by:

    1. Darja Rupnik Poklukar & Janez Žerovnik, 2023. "Double Roman Domination: A Survey," Mathematics, MDPI, vol. 11(2), pages 1-20, January.
    2. S. Banerjee & Michael A. Henning & D. Pradhan, 2020. "Algorithmic results on double Roman domination in graphs," Journal of Combinatorial Optimization, Springer, vol. 39(1), pages 90-114, January.

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