A Simple Proof of the Existence of a Planar Separator

Abstract: We provide a simple proof of the existence of a planar separator by showing that it is an easy consequence of the circle packing theorem. We also reprove other results on separators, including: (A) There is a simple cycle separator if the planar graph is triangulated. Furthermore, if each face has at most $d$ edges on its boundary, then there is a cycle separator of size O(sqrt{d n}). (B) For a set of n balls in R^d, that are k-ply, there is a separator, in the intersection graph of the balls, of size O(k^{1/d}n^{1-1/d}). (C) The k nearest neighbor graph of a set of n points in R^d contains a separator of size O(k^{1/d} n^{1-1/d}). The new proofs are (arguably) significantly simpler than previous proofs.