Central graphs and Italian domination parameters

Abstract

Abstract Let be a graph with and A function is said to be an Italian dominating function on a graph if every vertex with is adjacent to at least one vertex with or is adjacent to at least two vertices with . The value denotes the weight of an Italian dominating function. The minimum weight taken over all Italian dominating functions of is called Italian domination number and denoted by 
 Two parameters related to Italian dominating function are restrained Italian and total restrained dominating functions , for which the set of vertices with , and simultaneously the set of vertices with and the set of vertices with induce subgraphs with no isolated vertex respectively. The central graph of a graph is the graph obtained by subdividing each edge of exactly once and joining all the non-adjacent vertices of 
 In this work, we initiate the study of restrained (total restrained) Italian domination number of the central of any graph For a family of standard graphs we obtain the precise value of restrained (total restrained) Italian domination number for indeed for any graph G, the sharp bounds are provided for and for corona of , we establish the precise value of these parameters for