| Summary: | 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 γ{R3}(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.
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