Extract finite face data from a planar drawing

Open

WilliamsonShmoys.planar_drawing_face_data__p2m_87ce3ac9_d70a_4a22_9b10_c8d2544e53a0_a5

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A connected simple graph on at least three vertices with the given crossing-free drawing has a finite list of faces whose boundary lengths are at least three, sum to twice the number of edges, and satisfy Euler’s formula.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_face_data__p2m_87ce3ac9_d70a_4a22_9b10_c8d2544e53a0_a5
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (dH : DecidableRel H.Adj)
    (hconn : H.Connected)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hn : 3 ≤ Fintype.card V) :
    ∃ (m : ℕ) (faceLength : Fin m → ℕ),
      (∀ f, 3 ≤ faceLength f) ∧
      (∑ f : Fin m, faceLength f = 2 * H.edgeFinset.card) ∧
      (Fintype.card V + m = H.edgeFinset.card + 2) := by
  sorry

Source

David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).

View graph