A double Roman dominating function (DRDF) on a graph G=(V,E)\documentclass[12pt]{minimal}
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\begin{document}$$G=(V,E)$$\end{document} is a function f:V→{0,1,2,3}\documentclass[12pt]{minimal}
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\begin{document}$$f:V\rightarrow \{0,1,2,3\}$$\end{document} having the property that if f(v)=0\documentclass[12pt]{minimal}
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\begin{document}$$f(v)=0$$\end{document}, then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f(w)=3\documentclass[12pt]{minimal}
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\begin{document}$$f(w)=3$$\end{document}, and if f(v)=1\documentclass[12pt]{minimal}
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\begin{document}$$f(v)=1$$\end{document}, then vertex v must have at least one neighbor w with f(w)≥2\documentclass[12pt]{minimal}
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\begin{document}$$f(w)\ge 2$$\end{document}. The weight of a DRDF is the sum of its function values over all vertices, and the double Roman domination number γdR(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{dR}(G)$$\end{document} 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 C5\documentclass[12pt]{minimal}
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\begin{document}$$C_{5}$$\end{document} and C7\documentclass[12pt]{minimal}
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\begin{document}$$C_{7}$$\end{document}, then γdR(G)≤1110n.\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{dR}(G)\le \frac{11}{10}n.$$\end{document} Moreover, they presented an infinite family of graphs G\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {G}}$$\end{document} attaining the upper bound, and conjectured that G\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {G}}$$\end{document} 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.
