Let G = (V, E) be a finite simple graph where V = V (G) and E = E(G). Suppose that G has no isolated vertex. A covering total double Roman dominating func- tion (CT DRD function) f of G is a total double Roman dominating function (T DRD function) of G for which the set {v ∈ V (G)|f (v) 6 = 0} is a covering set. The covering total double Roman domination number γctdR(G) is the minimum weight of a CT DRD function on G. In this work, we present some contributions to the study of γctdR(G)- function of graphs. For the non star trees T , we show that γctdR(T ) ≤ 4n(T )+5s(T )−4l(T ) 3 , where n(T ), s(T ) and l(T ) are the order, the number of support vertices and the number of leaves of T respectively. Moreover, we characterize trees T achieve this bound. Then we study the upper bound of the 2-edge connected graphs and show that, for a 2-edge connected graphs G, γctdR(G) ≤ 4n 3 and finally, we show that, for a simple graph G of order n with δ(G) ≥ 2, γctdR(G) ≤ 4n 3 and this bound is sharp.
