A signed Italian dominating function on a graph \(G=(V,E)\) is a function \(f: V\rightarrow \{-1, 1, 2\}\) satisfying the condition that for every vertex u, \(f[u]\ge 1\). The weight of the signed Italian dominating function, f, is the value \(f(V)=\sum _{u\in V}f(u)\). The signed Italian dominating number of a graph G, denoted by \(\gamma _{sI}(G)\), is the minimum weight of a signed Italian dominating function on a graph G. In this paper, we prove that for any tree T of order \(n\ge 2\), \(\gamma _{sI}(T)\ge \frac{-n+4}{2}\) and we characterize all trees attaining this bound. In addition, we obtain some results about the signed Italian domination number of some graph operations. Furthermore, we prove that the signed Italian domination problem is \(\mathbf {NP}\)-Complete for bipartite graphs.

在图\（G =（V，E）\）上的带符号的意大利支配函数是满足以下条件的函数\（f：V \ rightarrow \ {-1，1，2 \} \）对于每个顶点u，\（f [u] \ ge 1 \）。带符号的意大利支配函数f的权重是值\（f（V）= \ sum _ {u \ in V} f（u）\）。图G的带符号的意大利支配数用\（\ gamma _ {sI}（G）\）表示，是图G上带符号的意大利支配函数的最小权重。在本文中，我们证明对于阶为\（n \ ge 2 \），\（\ gamma _ {sI}（T）\ ge \ frac {-n + 4} {2} \）的任何树T并且我们描述了达到此界限的所有树木的特征。此外，我们获得了一些图形操作的带符号的意大利支配数的一些结果。此外，我们证明了带符号的意大利控制问题是\（\ mathbf {NP} \） -二部图的完成。