An upper bound on the double Roman domination number

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 .