Extract Euler face data from a planar drawing

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WilliamsonShmoys.drawing_has_euler_face_data__p2m_55f47020_da90_4034_b4af_759a470f5d81_a2

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

approximation-algorithmsplanar-graphstreewidth

For the connected graph’s crossing-free drawing, produce a finite face count and a boundary length for each face. The data satisfy Euler’s equation, count each edge twice among face boundaries, and give every facial walk length at least three. The drawing predicate supplies the arcs; a proof must turn them into a plane embedding and establish its facial-walk properties. Connectedness, simplicity, and the vertex lower bound exclude short exceptional walks.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.drawing_has_euler_face_data__p2m_55f47020_da90_4034_b4af_759a470f5d81_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    (hconn : G.Connected)
    (hcard : 3 ≤ Fintype.card V) :
    letI : DecidableRel G.Adj := Classical.decRel G.Adj
    ∃ (faceCount : Nat) (boundaryLength : Fin faceCount → Nat),
      faceCount + Fintype.card V = G.edgeFinset.card + 2 ∧
      (∑ f : Fin faceCount, boundaryLength f) = 2 * G.edgeFinset.card ∧
      ∀ f, 3 ≤ boundaryLength f := 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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