Covering Italian domination in graphs

For a graph , an Italian dominating function (ID function) of is a function such that for each vertex with , , that is, either there is a vertex with or there are two vertices with . A function is a covering Italian dominating function (CID function) of if is an ID function and is a vertex cover set. The covering Italian domination number (CID number) is the minimum weight taken over all CID functions of .In this paper, we study the CID number in graphs. We show that the problem of computing this parameter is NP-hard even when restricted to some well-known families of graphs, and find some bounds on this parameter. We characterize the family of graphs for which their CID numbers attain the upper bound twice their vertex cover number as well as all claw-free graphs whose CID numbers attain the lower bound half of their orders. We also give the characterizations of some families of graphs with small CID numbers.