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 $D\_2\neq D\_1$?



Minimal dominating subsets in infinite graphs
Let $V(G)$ be the set of non-empty subsets of $\mathbb N$ and join two sets by an edge whenever they intersect. Let $D$ be the set of initial segments of $\mathbb N$. Then subsets of $D$ are dominating if and only if they are infinite.


