Quasi total double Roman domination in graphs

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 nonzero 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 γqtdR (G) equals the minimum weight of a QTDRD-function on G. In this paper, we first show that the problem of computing the quasi total double Roman domination number of a graph is NP-hard, and then we characterize graphs G with small or large γqtdR (G). Moreover, we establish an upper bound on the quasi total double Roman domination number and we characterize the connected graphs attaining this bound.