Suppose that $G=(V,E)$ is a simple, undirected graph. We say that $D\subseteq V$ is *dominating* if for all $v\in V\setminus D$ there is $d\in D$ such that $\{v,d\}\in E$. We say $D$ is *minimal dominating* if for all $d\in D$ we have that $D\setminus\{d\}$ is no longer dominating. There are graphs that [do not have minimal dominating sets](https://mathoverflow.net/questions/260931/minimal-dominating-subsets-in-infinite-graphs).


We call $G=(V,E)$ *flat* if for all $v\in V$ the neighborhood $N\_G(v)=\{w\in V:\{v,w\}\in E\}$ is finite.


If $G=(V,E)$ is a flat graph, does $G$ necessarily contain a minimal dominating set? Is it even the case that every dominating set contains a minimal dominating set? (Only answering the first question is sufficient for acceptance, but it would be lovely to have the answer to both questions.)



Minimal dominating sets in flat graphs
For flat graphs, dominating sets satisfy Zorn's condition: the intersection of a chain of dominating sets $D:=\cap D\_\alpha$ is dominating. Indeed, for any vertex $v$ the finite set $\{v\cup N(v)\}$ has non-empty intersection with every $D\_\alpha$, thus this finite non-empty intersection stabilizes and its limit is contained in $D$.


So, the answer is positive by Zorn lemma.


