On the total restrained double Italian domination

A double Italian  dominating (DID) function  of a graph $G=(V,E)$ is a function $f: V(G)\to\{0,1,2,3\}$ havingthe property that for every vertex $v\in V$, $\sum_{u\in N_G[v]}f(u)\geq 3$, if $f(v)\in \{0,1\}$.A restrained  double Italian dominating (RDID) function is a DID function $f$  such that the subgraph induced by the verticeswith label $0$ has no isolated vertex.A total restrained double Italian dominating (TRDID) function is an RDID function $f$  such that the set $\{v\in V: f(v)> 0\}$  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,$H \circ K_1$, $H \circ K_2$ and middle of them.