An Upper Bound on the Total Roman 2 -domination Number of Graphs with Minimum Degree Two

A total Roman \ (\2\\) -dominating function on a graph \ (G = (V, E) \) is a function \ (f: V\0, 1, 2\\) with the properties that (i) for every vertex \ (v V\) with \ (f (v) =0\), \ (f (N (v) ) 2\) and (ii) the set of vertices with \ (f (v) >0\) induces a subgraph with no isolated vertices. The weight of a total Roman \ (\2\\) -dominating function is the value \ (f (V) =ₕ Vf (v) \), and the minimum weight of a total Roman \ (\2\\) -dominating function is called the total Roman \ (\2\\) -domination number and denoted by \ (ₓₑ₂ (G) \). In this paper, we prove that for every graph \ (G\) of order \ (n\) with minimum degree at least two, \ (ₓₑ₂ (G) 5n6\).