An Italian dominating function (IDF) on a graph G is a function \(f:V(G)\rightarrow \{0,1,2\}\) such that for every vertex v with \(f(v)=0\), the total weight of f assigned to the neighbours of v is at least two, i.e., \(\sum _{u\in N_G(v)}f(u)\ge 2\). For any function \(f:V(G)\rightarrow \{0,1,2\}\) and any pair of adjacent vertices with \(f(v) = 0\) and u with \(f(u) > 0\), the function \(f_{u\rightarrow v}\) is defined by \(f_{u\rightarrow v}(v)=1\), \(f_{u\rightarrow v}(u)=f(u)-1\) and \(f_{u\rightarrow v}(x)=f(x)\) whenever \(x\in V(G){\setminus }\{u,v\}\). A secure Italian dominating function on a graph G is defined as an IDF f which satisfies that for every vertex v with \(f(v)=0\), there exists a neighbour u with \(f(u)>0\) such that \(f_{u\rightarrow v}\) is an IDF. The weight of f is \(\omega (f)=\sum _{v\in V(G) }f(v)\). The minimum weight among all secure Italian dominating functions on G is the secure Italian domination number of G. This paper is devoted to initiating the study of the secure Italian domination number of a graph. In particular, we prove that the problem of finding this parameter is NP-hard and we obtain general bounds on it. Moreover, for certain classes of graphs, we obtain closed formulas for this novel parameter.

图G上的意大利语支配函数（IDF）是函数\（f：V（G）\ rightarrow \ {0,1,2 \} \），使得对于每个具有\（f（v）= 0的顶点v \），分配给v的邻居的f的总权重至少为2，即\（\ sum _ {u \ in N_G（v）} f（u）\ ge 2 \）。对于任何函数\（f：V（G）\ rightarrow \ {0,1,2 \} \）和任意一对相邻顶点均为\（f（v）= 0 \）和u为\（f（u） > 0 \），函数\（f_ {u \ rightarrow v} \）由\（f_ {u \ rightarrow v}（v）= 1 \），\（f_ {u \ rightarrow v}（u）定义= f（u）-1 \）和\（f_ {u \ rightarrow v}（x）= f（x）\）每当\（x \ in V（G）{\ setminus} \ {u，v \} \）时。在图上一个安全意大利控制函数G ^定义为IDF ˚F其满足，对于每一个顶点v与\（F（V）= 0 \） ，存在邻居ù与\（F（U）> 0 \）这样\（f_ {u \ rightarrow v} \）是IDF。f的权重为\（\ omega（f）= \ sum _ {v \ in V（G）} f（v）\）。在所有安全意大利主导功能之间的最小重量摹是安全意大利控制数g ^。本文致力于启动图的安全意大利支配数的研究。特别地，我们证明了找到此参数的问题是NP-难的，并且我们获得了关于它的一般边界。此外，对于某些类的图，我们获得了该新颖参数的封闭公式。