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.