Let 𝐺 = (𝑉,𝐸) be a graph with 𝑉 = 𝑉 (𝐺) and 𝐸 = 𝐸(𝐺). A function 𝑓∶ 𝑉 → {0,1,2} is said to be an Italian dominating function on a graph 𝐺 if every vertex 𝑢 with 𝑓 (𝑢)=0 is adjacent to at least one vertex 𝑣 with 𝑓 (𝑣)=2 or is adjacent to at least two vertices 𝑥,𝑦 with 𝑓 (𝑥) = 𝑓 (𝑦) = 1. The value 𝑤(𝑓 ) = Σ𝑓 (𝑣)𝑣∈𝑉 denotes the weight of an Italian dominating function. The minimum weight taken over all Italian dominating functions of 𝐺 is called Italian domination number and denoted by 𝛾𝐼(𝐺). Two parameters related to Italian dominating function (IDF) are restrained Italian (RIDF) and total restrained (TRDF) dominating functions 𝑓 , for which the set of vertices 𝑣 with 𝑓 (𝑣) = 0, and simultaneously the set of vertices 𝑣 with 𝑓 (𝑣) > 0 and the set of vertices 𝑣 with 𝑓 (𝑣)= 0 induce subgraphs with no isolated vertex respectively. The central graph 𝐶(𝐺) of a graph 𝐺 is the graph obtained by subdividing each edge of 𝐺 exactly once and joining all the non-adjacent vertices of 𝐺. In this work, we initiate the study of restrained (total restrained) Italian domination number of the central of any graph 𝐺. For a family of standard graphs 𝐺, we obtain the precise value of restrained (total restrained) Italian domination number for 𝐶(𝐺), indeed for any graph G, the sharp bounds are provided for 𝐶(𝐺), and for 𝐺 corona of 𝐾1 , 𝐺∘𝐾1 we establish the precise value of these parameters for 𝐶(𝐺∘𝐾1).
