A Simple Proof of the Existence of a Planar Separator

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 ` ? dn ̆ . (B) For a set of n balls in Rd, that are k-ply, there is a separator, in the intersection graph of the balls, of size O ` k1{dn1 ́1{d ̆ . (C) The k nearest neighbor graph of a set of n points in Rd contains a separator of size O ` k1{dn1 ́1{d ̆ . The new proofs are (arguably) significantly1 simpler than previous proofs.