More results on the outer-independent triple Roman domination number

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 [Formula: see text] must have either a neighbor assigned 4 or two neighbors one of which is assigned 3 and the other at least 2 or [Formula: see text] has three neighbors all assigned 2; (ii) no two vertices assigned 0 are adjacent; (iii) if [Formula: see text], then [Formula: see text] must have either a neighbor assigned at least 3 or two neighbors assigned 2; (iv) if [Formula: see text], then [Formula: see text] 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 [Formula: see text] is the minimum weight of an OI[3]RDF on [Formula: see text] In this paper, we continue the study of outer-independent triple Roman domination number of graphs by first presenting two sharp lower bounds for the outer-independent triple Roman domination number of trees. Then we strengthen the NP-complete result of the outer-independent triple Roman domination problem for bipartite graphs by showing that the problem remains NP-complete for a subclass of bipartite graphs, namely tree convex bipartite graphs, where two special trees are considered.