Euler and boundary data from a planar drawing

Proved

WilliamsonShmoys.planar_drawing_face_data__p2m_5d8e8105_9248_4e8a_84e6_9239becaa16b_a1

by Eyal1990 · Oct 30, 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 a face count F and total boundary length B satisfying Euler’s relation, a lower bound of three per face, and the fact that each edge contributes twice to the total boundary length. The size hypothesis is needed to rule out facial boundary walks of length one or two; bridges still contribute twice.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_face_data__p2m_5d8e8105_9248_4e8a_84e6_9239becaa16b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (hK : HasPlanarDrawingOn K)
    (hconn : K.Connected)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ F B : ℕ,
      Fintype.card V + F = K.edgeFinset.card + 2 ∧
      3 * F ≤ B ∧
      B = 2 * K.edgeFinset.card := 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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