More results on the outer-independent triple Roman domination number.

An outer-independent triple Roman dominating function (OI[3]RDF) on a graph G = (V,E) is function f : V →{0, 1, 2, 3, 4} having the property that (i) if f(v) = 0, 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 f(v) = 1, then v must have either a neighbor assigned at least 3 or two neighbors assigned 2; (iv) if f(v) = 2, 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 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. [ABSTRACT FROM AUTHOR]