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 γdR(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 γdR(G)≤6n5 and posed the question whether this bound can be improved. Amjadi et al. (2018) settled this question by proving that γdR(G)≤8n7. In this paper, we improve this bound to γdR(G)≤11n10. Moreover, we provide an infinite family of graphs attaining this bound.