An upper bound on the double Roman domination number

A double Roman dominating function (DRDF) on a graph is a function having the property that if , then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with , and if , then vertex v must have at least one neighbor w with . The weight of a DRDF f is the value . The double Roman domination number 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 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, .