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.