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A Simple Proof of the Existence of a Planar Separator

Sariel Har-Peled

TL;DR

This paper provides a simple, self-contained proof of the planar separator theorem by deriving it from Koebe's circle packing theorem. The core idea is to realize a planar graph as a kissing graph of interior-disjoint disks and to form a separator from disks intersecting a randomly chosen circle in the packing, yielding a separator of size $O(\sqrt{n})$ that leaves components of size at most $(9/10)n$. It further extends the approach to weighted graphs, cycle separators (for triangulated and non-triangulated graphs), and multiple geometric graph settings including $k$-ply ball systems and kth-nearest-neighbor graphs, with corresponding separator bounds such as $4 \sqrt{n}$ or $O(k^{1/d} n^{1-1/d})$. The results collectively demonstrate that circle packing provides a simple and versatile framework for deriving various separators, with implications for divide-and-conquer algorithms on planar graphs and related geometric graphs. An open question remains whether there exists a finite algorithm to compute circle packings that realize planar graphs, or at least packings with bounded overlap, to enable practical separator computations.

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.

A Simple Proof of the Existence of a Planar Separator

TL;DR

This paper provides a simple, self-contained proof of the planar separator theorem by deriving it from Koebe's circle packing theorem. The core idea is to realize a planar graph as a kissing graph of interior-disjoint disks and to form a separator from disks intersecting a randomly chosen circle in the packing, yielding a separator of size that leaves components of size at most . It further extends the approach to weighted graphs, cycle separators (for triangulated and non-triangulated graphs), and multiple geometric graph settings including -ply ball systems and kth-nearest-neighbor graphs, with corresponding separator bounds such as or . The results collectively demonstrate that circle packing provides a simple and versatile framework for deriving various separators, with implications for divide-and-conquer algorithms on planar graphs and related geometric graphs. An open question remains whether there exists a finite algorithm to compute circle packings that realize planar graphs, or at least packings with bounded overlap, to enable practical separator computations.

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 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.

Paper Structure

This paper contains 15 sections, 10 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of the planar separator theorem
  3. The proof
  4. Remarks.
  5. Extensions
  6. Weighted version
  7. Cycle separators
  8. Cycle separator if the graph is not triangulated.
  9. Ball systems that are k-ply
  10. Separators for the kth nearest neighbor graph
  11. Separator for r vertices in a planar graph
  12. Conclusions
  13. An open problem.
  14. Acknowledgments
  15. Proof of Lemma 2.2 with a better constant

Key Result

Lemma 2.1

The separator $S$ breaks ${G}$ into two subgraphs with at most $(9/10) n$ vertices in each connected component.

Figures (3)

  • Figure 2.1:
  • Figure 2.2: How to cover a disk of radius $2$ by $7$ disks of radius $1$.
  • Figure 3.1:

Theorems & Definitions (11)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Definition 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Lemma 3.7
  • ...and 1 more