An improved upper bound on the double Roman domination number of graphs with minimum degree at least two

Abstract 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 is the sum of its function values over all vertices, and the double Roman domination number γ d R ( G ) is the minimum weight of a DRDF on G . Let G be a connected graph G of order n and minimum degree at least two. With the exception of seven graphs of order at most seven, Beeler et al. (2016) observed that γ d R ( G ) ≤ 6 n 5 and posed the question whether this bound can be improved. Amjadi et al. (2018) settled this question by proving that γ d R ( G ) ≤ 8 n 7 . In this paper, we improve this bound to γ d R ( G ) ≤ 11 n 10 . Moreover, we provide an infinite family of graphs attaining this bound.