On computing total double Roman domination number of trees in linear time

Let G = (V;E) be a graph. A double Roman dominating function (DRDF) on G is a function f : V ! f0; 1; 2; 3g such that for every vertex v 2 V if f(v) = 0, then either there is a vertex u adjacent to v with f(u) = 3 or there are vertices x and y adjacent to v with f(x) = f(y) = 2 and if f(v) = 1, then there is a vertex u adjacent to v with f(u) 2. A DRDF f on G is a total DRDF (TDRDF) if for any v 2 V with f(v) > 0 there is a vertex u adjacent to v with f(u) > 0. The weight of f is the sum f(V ) = P v2V f(v). The minimum weight of a TDRDF on G is the total double Roman domination number of G. In this paper, we give a linear algorithm to compute the total double Roman domination number of a given tree.