Further results on outer independent triple Roman domination

An outer-independent triple Roman dominating function (OI[3]RDF) on a graph [Formula: see text] is function [Formula: see text] having the property that (i) if [Formula: see text] then v must have either a neighbor assigned 4 or two neighbors one of which is assigned 3 and the other at least 2 or v has three neighbors all assigned 2; (ii) no two vertices assigned 0 are adjacent; (iii) if [Formula: see text], then v must have either a neighbor assigned at least 3 or two neighbors assigned 2; (iv) if [Formula: see text], then v must have one neighbor assigned at least 2. The weight of an OI[3]RDF is the sum of its function value over the whole set of vertices, and the outer-independent triple Roman domination number of G is the minimum weight of an OI[3]RDF on G. In this paper, we continue the study of outer-independent triple Roman domination number in graphs. First, we characterize all graphs with small or large outer-independent triple Roman domination number, and then we establish relationships with some other related parameters. Finally we present a sharp upper bound for the outer-independent triple Roman domination number of trees.