Disprove of a conjecture on the double Roman domination number

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 _dR(G) is the minimum weight of a DRDF on G . Khoeilar et al. (Discrete Appl. Math. 270:159–167, 2019) proved that if G is a connected graph of order n with minimum degree two different from C_5 and C_7 , then γ _dR(G)≤11/10n. Moreover, they presented an infinite family of graphs 𝒢 attaining the upper bound, and conjectured that 𝒢 is the only family of extremal graphs reaching the bound. In this paper, we disprove this conjecture by characterizing all extremal graphs for this bound.