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 \(f(V)=∑_{{v}∈ V}f({v})\), and the minimum weight of a total Roman \(\{2\}\)-dominating function is called the total Roman \(\{2\}\)-domination number and denoted by \(γtR2(G)\). In this paper, we prove that for every graph \(G\) of order \(n\) with minimum degree at least two, \(γtR2({G})≤ 5n/6\).