Restrained double Italian domination in graphs

Let G be a graph with vertex set V (G). A double Italian dominating function (DIDF) is a function f : V (G) -> {0, 1, 2, 3} having the property that f(N[u]) >= 3 for every vertex u is an element of V (G) with f(u) is an element of {0, 1}, where N[u] is the closed neighborhood of u. If f is a DIDF on G, then let V-0 = {v is an element of V (G) : f(v) = 0}. A restrained double Italian dominating function (RDIDF) is a double Italian dominating function f having the property that the subgraph induced by V-0 does not have an isolated vertex. The weight of an RDIDF f is the sum Sigma(v is an element of(G)) f(v), and the minimum weight of an RDIDF on a graph G is the restrained double Italian domination number. We present bounds and Nordhaus-Gaddum type results for the restrained double Italian domination number. In addition, we determine the restrained double Italian domination number for some families of graphs.