For a graph G with no isolated vertex, a covering total double Roman dominating function (CTDRD function) f of G is a total double Roman dominating function (TDRD function) of G for which the set fv 2 V (G)jf(v) 6= 0g is a vertex cover set. The covering total double Roman domination number ctdR(G) equals the minimum weight of an CTDRD function on G. An CTDRD function on G with weight ctdR(G) is called a ctdR(G)-function. In this paper, the graphs G with small ctdR(G) are characterised. We show that the decision problem associated with CTDRD is NP- complete even when restricted to planer graphs with maximum degree at most four. We then show that for every graph G without isolated vertices, oitR(G) < ctdR(G) < 2 oitR(G) and for every tree T, 2 (T) + 1 ctdR(T) 4 (T), where oitR(G) and (T) are the outer independent total Roman domination number of G, and the minimum vertex cover number of T respectively. Moreover we investigate the ctdR of corona of two graphs.
