A graph is outerplanar if it is planar and can be embedded in the plane so that all vertices are in the boundary of the outer face.
The analogue of Kuratowski's theorem for outerplanar graphs is that a graph is outerplanar if and only if it has no $K_4$ or $K_{2,3}$ minors. The usual way to show that a graph is not planar is via Euler's formula. I am wondering if there is a clean way to show that $K_4$ and $K_{2,3}$ are not outerplanar.