New bounds on the outer-independent total double Roman domination number

Abstract

A double Roman dominating function (DRDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying (i) if [Formula: see text] then there must be at least two neighbors assigned two under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text]; and (ii) if [Formula: see text] then [Formula: see text] must be adjacent to a vertex [Formula: see text] such that [Formula: see text]. A DRDF is an outer-independent total double Roman dominating function (OITDRDF) on [Formula: see text] if the set of vertices labeled [Formula: see text] induces an edgeless subgraph and the subgraph induced by the vertices with a non-zero label has no isolated vertices. The weight of an OITDRDF is the sum of its function values over all vertices, and the outer-independent total Roman domination number [Formula: see text] is the minimum weight of an OITDRDF on [Formula: see text]. In this paper, we establish various bounds on [Formula: see text]. In particular, we present Nordhaus–Gaddum-type inequalities for this parameter. Some of our results improve the previous results.