An upper bound on the double Roman domination number

A double Roman dominating function (DRDF) on a graph G = ( V , E ) is a function f : V → { 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 ) ≥ 2 . The weight of a DRDF f is the value f ( V ) = ∑ u ∈ V f ( u ) . The double Roman domination number 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 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, γ dR ( G ) ≤ 8 n 7 .