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∈Vf(u). The double Roman domination number$$\gamma _{dR}(G)$$γdR(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 $$\gamma _{dR}(G)\le \frac{6n}{5}$$γdR(G)≤6n5 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, $$\gamma _{dR}(G)\le \frac{8n}{7}$$γdR(G)≤8n7.