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∈V(G) with f(u)∈{0,1}, where N[u] is the closed neighborhood of u. If f is a DIDF on G, then let V0={v∈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 V0 does not have an isolated vertex. The weight of an RDIDF f is the sum ∑v∈V(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.