On the outer-independent total (Roman) domination number of some graph operators

The goal of this article is to obtain closed formulas for the outer-independent total domination number and the outer-independent total Roman domination number of the following well-known graph operators defined from a connected graph G: the central graph C(G), the middle graph M(G), the graph operator R(G) and the total graph T(G). We show that these formulas can be obtained for the last three graph operators and for the outer-independent total domination number of C(G). The picture is quite different when it concerns the outer-independent total Roman domination number of C(G). In this case, we obtain tight bounds and, imposing some restrictions on G, we obtain closed formulas.