Counting Weight-$k$ Italian Dominating Sets on Trees and Cycles

We study the problem of counting \emph{Italian dominating functions} (IDFs) of each possible weight on paths, trees, and cycles. An Italian dominating function on a graph $G=(V,E)$ is a function $f\colon V\to\{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ satisfies $\sum_{u\in N(v)}f(u)\ge 2$; t