D. A. Mojdeh - Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
M. Abdallah - Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

A double Italian  dominating (DID) function  of a graph G=(V,E) is a function f: V(G)\to\{۰,۱,۲,۳\} havingthe property that for every vertex v\in V, \sum_{u\in N_G[v]}f(u)\geq ۳, if f(v)\in \{۰,۱\}.A restrained  double Italian dominating (RDID) function is a DID function f  such that the subgraph induced by the verticeswith label ۰ has no isolated vertex.A total restrained double Italian dominating (TRDID) function is an RDID function f  such that the set \{v\in V: f(v)> ۰\}  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 \circ K_۱, H \circ K_۲ and middle of them.

Double Italian domination, (total) restrained double Italian domination, middle graph