Remarks on the restrained Italian domination number in graphs

Let G be a graph with vertex set V (G). An Italian dominating function (IDF) is a function f : V (G) -> {0, 1, 2} having the property that that f(N(u)) >= 2 for every vertex u is an element of V (G) with f(u) = 0, where N(u) is the neighborhood of u. If f is an IDF on G, then let V-0 = {v is an element of V (G) : f(v) = 0}. A restrained Italian dominating function (RIDF) is an Italian dominating function f having the property that the subgraph induced by V-0 does not have an isolated vertex. The weight of an RIDF f is the sum Sigma(v is an element of 2V(G)) f(v), and the minimum weight of an RIDF on a graph G is the restrained Italian domination number. We present sharp bounds for the restrained Italian domination number, and we determine the restrained Italian domination number for some families of graphs.