Geodesic
A New Upper Bound for the Perfect Italian Domination Number of a Tree
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 3, pp. 1005-1022

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A perfect Italian dominating function (PIDF) on a graph G is a function f : V (G) →{ 0, 1, 2 } satisfying the condition that for every vertex u with f(u) = 0, the total weight of f assigned to the neighbors of u is exactly two. The weight of a PIDF is the sum of its functions values over all vertices. The perfect Italian domination number of G, denoted γ_I^p (G), is the minimum weight of a PIDF of G. In this paper, we show that for every tree T of order n ≥ 3, with 𝓁 (T) leaves and s(T) support vertices, γ_I^p (T) ≥4n- 𝓁(T) + 2s (T) - 15, improving a previous bound given by T.W. Haynes and M.A. Henning in [Perfect Italian domination in trees, Discrete Appl. Math. 260 (2019) 164–177].
Keywords: Italian domination, Roman domination, perfect Italian domination 
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Nazari-Moghaddam, Sakineh; Chellali, Mustapha. A New Upper Bound for the Perfect Italian Domination Number of a Tree. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 3, pp. 1005-1022. http://geodesic.mathdoc.fr/item/DMGT_2022_42_3_a19/