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.