An Italian dominating function on a graph G with vertex set V(G) is a function f:V(G)→{0,1,2}\documentclass[12pt]{minimal}
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\begin{document}$$f :V(G) \rightarrow \{0,1,2\}$$\end{document} having the property that for every vertex v with f(v)=0\documentclass[12pt]{minimal}
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\begin{document}$$f(v) = 0$$\end{document}, at least two neighbors of v are assigned 1 under f or at least one neighbor of v is assigned 2 under f. The weight of an Italian dominating function f is the sum of the values assigned to all the vertices under f. The Italian domination number of G, denoted by γI(G)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{I}(G)$$\end{document}, is the minimum weight of an Italian dominating of G. It is known that if G is a connected graph of order n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 3$$\end{document}, then γI(G)≤34n\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{I}(G) \le \frac{3}{4}n$$\end{document}. Further, if G has minimum degree at least 2, then γI(G)≤23n\documentclass[12pt]{minimal}
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\begin{document}$$\gamma _{I}(G) \le \frac{2}{3}n$$\end{document}. In this paper, we characterize the connected graphs achieving equality in these bounds. In addition, we prove Nordhaus–Gaddum inequalities for the Italian domination number.
