Volume 11 • Issue 1 • PP: 35-46 • 2024
On the Perfect Italian Domination Numbers of Some Graph Classes
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© 2024 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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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.
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