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∈V(G) with f(u)=0, where N(u) is the neighborhood of u. If f is an IDF on G, then let V0={v∈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 V0 does not have an isolated vertex. The weight of an RIDF f is the sum ∑v∈V(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.