Planar Graphs with Separation Are dp-Minimal

We prove that, given a planar embedding of a graph in the sphere, the expansion of the graph structure by predicates encoding vertex separation by simple graph cycles is dp-minimal. This provides a rich natural class of examples of unstable, dp-minimal, and also monadically NIP theories. We also show how to infer the existence of a distal expansion of the theory of the Farey graph.