For a graphG=(V,E)withV=V(G)andE=E(G), a perfect double Italian dominating functionis a functionf:V→{0, 1, 2, 3}having the property that 3≤∑u∈NG[v]f(u)≤4, for every vertexv∈Gwithf(v)∈{0, 1}. The weight of a perfect double Italian dominating functionfis the sumf(V)=∑v∈V(G)f(v)and the minimum weight of a perfect double Italian dominating function onGis theperfect double Italian domination numberγpdI(G)ofG. We initiate the study of perfect double Italiandominating functions. We check theγpdIof some standard graphs and evaluate withγdIof such graphs.The perfect double Italian dominating functions versus perfect double Roman dominating functionsare perused. The NP-completeness of this parameter is verified even when it is restricted to bipartitegraphs. Finally, we characterize the graphsGof ordernwithγpdI(G)∈{3, 4, 5,n,2n−3, 2n−4, 2n−5, 2n−6}.
