Disprove of a conjecture on the double Roman domination number

A double Roman dominating function (DRDF) on a graph \(G=(V,E)\) is a function \(f:V\rightarrow \{0,1,2,3\}\) 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 \(f(w)\ge 2\). The weight of a DRDF is the sum of its function values over all vertices, and the double Roman domination number \(\gamma _{dR}(G)\) 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 \(\gamma _{dR}(G)\le \frac{11}{10}n.\) Moreover, they presented an infinite family of graphs \({\mathcal {G}}\) attaining the upper bound, and conjectured that \({\mathcal {G}}\) 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.