Restrained Italian bondage number in graphs

A restrained Italian dominating function (RIDF) on a graph G = (V, E) is a function f : V -> {0, 1, 2} satisfying: (i) f (N (u)) >= 2 for every vertex u is an element of V (G) with f (u) = 0, where N (u) is the set of vertices adjacent to u; (ii) the subgraph induced by the vertices assigned 0 under f has no isolated vertices. The weight of an RIDF is the sum of its function values over the whole set of vertices, and the restrained Italian domination number gamma(rI) (G) is the minimum weight of an RIDF on G. In this paper, we initiate the study of the restrained Italian bondage number b(rI) (G) of a graph G with no isolated vertices defined as the smallest size of set of edges F subset of E(G) for which gamma(rI) (G - F) > gamma(rI) (G). We begin by showing that the decision problem associated with the restrained Italian bondage problem is NP-hard. Then basic properties of the restrained Italian bondage number are presented. Finally, some sharp bounds for b(rI) (G) are also established.