Let $G$ be a graph with no cycles. We want to either prove or disprove that any minor graph of $G$ is either acyclic or cyclic.
My idea:
If $G$ is cyclic, all connected components are trees. Choose one of these connected components, say $C$. A minor of $C$ can be obtained with vertex and edge deletions as well as edge contractions. Vertex and edge deletions cannot add any cycles. And I am quite sure edge contractions cannot do so either, but here I might be mistaken. Is it possible for edge contractions to add cycles to the component $C$?
We assume $G$ is finite.