Let $G=(V,E)$ be a finite, simple, undirected graph. We call $D\subseteq V$ *cycle-intersecting* if for every [simple cycle](https://en.wikipedia.org/wiki/Cycle_(graph_theory)) $C\subseteq V$ we have $C\cap D \neq \emptyset$.


Is there a graph $G$ such that for every cycle-intersecting subset $D$ and for every simple cycle $C$ we have $|C\cap D| > 1$?



Cycle-intersecting subsets
No. And this is not about cycles at all.


Take minimal cycle-intersecting subset. If it has at least two vertices from any cycle, remove any vertex from it and get a smaller cycle-intersecting subset. 


