Euler and face-incidence certificate for a planar drawing

Proved

WilliamsonShmoys.planar_face_count_certificate__p2m_961b854d_f4cc_4b36_ad05_39668b153494_a2

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

approximation-algorithmsplanar-graphstreewidth

For a finite connected simple graph with at least three vertices and a crossing-free drawing, there are finitely many facial boundary walks whose count satisfies Euler’s formula, whose total length counts every edge twice, and each of whose lengths is at least three.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_face_count_certificate__p2m_961b854d_f4cc_4b36_ad05_39668b153494_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (hV : 3 ≤ Fintype.card V)
    (hconn : K.Connected)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    ∃ (F : Nat) (faceLength : Fin F → Nat),
      Fintype.card V + F = K.edgeFinset.card + 2 ∧
      (∑ i : Fin F, faceLength i) = 2 * K.edgeFinset.card ∧
      ∀ i, 3 ≤ faceLength i := 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).

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