A double Italian dominating (DID) function of a graph G = (V; E) is a function f : V (G) ! f0; 1; 2; 3g having the property that for every vertex v 2 V , Pu2NG[v] f(u) ≥ 3, if f(v) 2 f0; 1g. A restrained double Italian dominating (RDID) function is a DID function f such that the subgraph induced by the vertices with label 0 has no isolated vertex. A total restrained double Italian dominating (TRDID) function is an RDID function f such that the set fv 2 V : f(v) > 0g induces a subgraph with no isolated vertex. We initiate the study of TRDID function of any graph G. The TRDID and RDID functions of the middle of any graph G are investigated, and then, the sharp bounds for these parameters are established. Finally, for a graph H, we provide the minimum value of TRDID and RDID functions for corona graphs, H ◦ K1, H ◦ K2 and middle of them.
