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) having the property that f(N[u]) 3 for every vertex u 2 V (G) with f(u) 2 f0; 1g, where N[u] is the closed neighborhood of u. If f is a DIDF on G, then let V0 = fv 2 V (G): f(v) = 0g. A restrained double Ital-ian dominating function (RDIDF) is a double Italian dominating function f having the property that the subgraph induced by V0 does not have an isolated vertex. The weight of an RDIDF f is the sum P v2V (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. [ABSTRACT FROM AUTHOR]