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 → { 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 10 n . 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.