An Italian dominating function on a graph G = (V, E) is defined as a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2 or at least two vertices x, y for which f(x) = f(y) = 1. The weight of an Italian dominating function is w(f) = Σv∈V f(v). The Italian domination number is the minimum weight taken over all Italian dominating functions of G and denoted by γI(G). Three domination parameters related to the Italian dominating function are total Italian, restrained Italian, and total restrained Italian dominating function. A total ((restrained) (total restrained)) Italian dominating function f is an Italian dominating function if the set of vertices with positive label ((the set of vertices with label 0), (at the same time, the set of vertices with positive label and the set of vertices with label 0)) induces ((induces) (induce)) a subgraph with no isolated vertex. A minimum weight of any total ((restrained) (total restrained)) Italian dominating function f is called a total ((restrained) (total restrained)) Italian domination number denoted by γtI(G), ((γrI(G)) (γtrI(G))). We initiate the study of parameters, restrained and total restrained Italian domination number of a graph G and the middle graph of G. For the family of standard graphs, we obtain the exact value of these parameters. For arbitrary graph G, we obtain the sharp bounds of these parameters, and for some corona graphs, we establish the precise value of these parameters.
