Every graph has a planar covering graph. In particular, the Universal cover, being a tree, will be planar.
How do we find a minimal planar covering $P$ graph of a graph $G$? By minimal, I mean that the only planar covering graph of $G$ that $P$ covers is $P$ itself.
I conjecture that it will involve replacing all $K_5$ and $K_{3,3}$ minors with their minimal planar covering graphs, but I'm not sure.
Notes: If $G$ is planar, then $P = G$. If $G$ has a planar cover (a finite planar covering graph), then $P$ will be a planar cover.