Journal of Combinatorial Optimization

, Volume 36, Issue 1, pp 81–89 | Cite as

An upper bound on the double Roman domination number

  • J. Amjadi
  • S. Nazari-Moghaddam
  • S. M. SheikholeslamiEmail author
  • L. Volkmann


A double Roman dominating function (DRDF) on a graph \(G=(V,E)\) is a function \(f : V \rightarrow \{0, 1, 2, 3\}\) having the property that if \(f(v) = 0\), then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with \(f(w)=3\), and if \(f(v)=1\), then vertex v must have at least one neighbor w with \(f(w)\ge 2\). The weight of a DRDF f is the value \(f(V) = \sum _{u \in V}f(u)\). The double Roman domination number \(\gamma _{dR}(G)\) 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 \(\gamma _{dR}(G)\le \frac{6n}{5}\) 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, \(\gamma _{dR}(G)\le \frac{8n}{7}\).


Double Roman domination number Roman domination Domination 


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsAzarbaijan Shahid Madani UniversityTabrizIslamic Republic of Iran
  2. 2.Lehrstuhl II für MathematikRWTH Aachen UniversityAachenGermany

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