Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 10.54216/GJSMA https://www.americaspg.com/journals/show/3181 2022 2022 Umm Al-Qura University, Mekka, Saudi Arabia Khadija Khadija A function f:V(G)→{0,1,2} is called a Perfect Italian dominating function (PIDF) of a graph G=(V,E) if ∑_(v∈N(u)) f(v)=2 for every vertex u∈V(G) with f(u)=0. The weight of an PIDF is w(f)=∑_(v∈V) f(v) . The minimum weight of all Perfect Italian dominating functions that can be conducted on a graph G is called the perfect Italian domination number of G and is denoted by γ_I^p (G). In this paper, we study the problem on different graph classes. We determine the perfect Italian domination numbers of the circulant graphs C_n {1,2} for n≥5 and give upper bounds for γ_I^p (C_n {1,3})  when n≥7. We also find this parameter for generalized Petersen graph P(n,2) when n≥5. We determine γ_I^p (G) of strong grids P_2⊠P_n  and P_3⊠P_n for arbitrary n≥2, then we introduce an upper bound for γ_I^p (P_m⊠P_n ) when m,n≥2 are arbitraries. Finally, we determine the perfect Italian domination number of Jahangir graph J_(s,m) for arbitrary s≥2 and m≥3. 2024 2024 35 46 10.54216/GJMSA.0110104 https://www.americaspg.com/articleinfo/33/show/3181