An Italian dominating function on a graph $G$ is a function $f:V(G)\rightarrow \{0,1,2\}$ such that $\sum_{u\in N(v)}f(u)\geq 2$ for every vertex $v\in V_0$, where $V_0=\{v\in V(G) : f(v)=0\}$ and $N(v)$ represents the open neighbourhood of $v$. A starred Italian dominating function on $G$ is an Italian dominating function $f$ such that $V_0$ is not a dominating set of $G$. The starred Italian domination number of $G$, denoted $\gamma_{I}^*(G)$, is the minimum weight $\omega(f)=\sum_{v\in V(G)}f(v)$ among...