Bounds on the quasi-total double Roman domination number in graphs

A quasi-total double Roman dominating function (QTDRD-function) on a graph [Formula: see text] is a function [Formula: see text] having the property that (i) if [Formula: see text], then vertex [Formula: see text] must have at least two neighbors assigned 2 under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text]; (ii) if [Formula: see text], then vertex [Formula: see text] has at least one neighbor [Formula: see text] with [Formula: see text], and (iii) if [Formula: see text] is an isolated vertex in the subgraph induced by the set of vertices assigned nonzero values, then [Formula: see text]. The weight of a QTDRD-function [Formula: see text] is the sum of its function values over the whole vertices, and the quasi-total double Roman domination number [Formula: see text] equals the minimum weight of a QTDRD-function on [Formula: see text]. In this paper, we first provide some Nordhaus–Gaddum-type results for the sum [Formula: see text] where [Formula: see text] is the complement graph of [Formula: see text] Then sharp bounds involving the quasi-total double Roman domination number with some domination parameters are established. Restricted to the class of trees [Formula: see text] we present a lower bound on [Formula: see text] in terms of the order and the numbers of leaves and support vertices.