Let $G=(V,E)$ be any simple, undirected graph. A *dominating set* is a set $D\subseteq V$ such that for all $v\in V\setminus D$ there is $d\in D$ such that $\{v,d\}\in E$.

Is there an infinite graph $G=(V,E)$ such that there is a dominating subset $D\subseteq V$ such that for any dominating subset $D_1\subseteq D$ there is a dominating subset $D_2\subseteq D_1$ with $D_2\neq D_1$?