Quasi total double Roman domination in trees

A quasi total double Roman dominating function (QTDRD-function) on a graph G = (V (G), E(G)) is a function f : V (G) −→ {0, 1, 2, 3} having the property that (i) 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; (ii) if f(v) = 1, then vertex v has at least one neighbor w with f(w) ≥ 2, and (iii) if x is an isolated vertex in the subgraph induced by the set of vertices assigned non-zero values, then f(x) = 2. The weight of a QTDRD-function f is the sum of its function values over the whole vertices, and the quasi total double Roman domination number γ_qtdR(G) equals the minimum weight of a QTDRD-function on G. In this paper, we show that for any tree T of order n ≥ 4, γ_qtdR(T) ≤ n+ ^s(₂^T) , where s(T) is the number of support vertices of T, that improves a known bound.