Geodesic
Bounds on the Double Italian Domination Number of a Graph
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1129-1137.

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For a graph G, a Roman 3-dominating function is a function f : V → 0, 1, 2, 3 having the property that for every vertex u ∈ V, if f(u) ∈ 0, 1, then f(N[u]) ≥ 3. The weight of a Roman 3-dominating function is the sum w(f) = f(V) = Σv∈V f(v), and the minimum weight of a Roman 3-dominating function is the Roman 3-domination number, denoted by γR3(G). In this paper, we present a sharp lower bound for the double Italian domination number of a graph, and improve previous bounds given in [D.A. Mojdeh and L. Volkmann, Roman 3-domination (double Italian domination), Discrete Appl. Math. 283 (2022) 555–564]. We also present a probabilistic upper bound for a generalized version of double Italian domination number of a graph, and show that the given bound is asymptotically best possible.
Keywords: Italian domination, double Italian domination, probabilistic methods 
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Azvin, Farzaneh; Rad, Nader Jafari. Bounds on the Double Italian Domination Number of a Graph. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1129-1137. http://geodesic.mathdoc.fr/item/DMGT_2022_42_4_a7/

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