An upper bound on the double Roman domination number

Abstract

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, $$\gamma _{dR}(G)\le \frac{8n}{7}$$ .