Euler and face count from a connected planar drawing

Proved

WilliamsonShmoys.connected_drawing_face_count__p2m_044c324d_acb7_4615_b70a_8b8b8a044876_a2

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every crossing-free drawing of a finite connected simple graph with at least three vertices yields a natural number of faces satisfying Euler's equation and the lower bound on total facial boundary length.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.connected_drawing_face_count__p2m_044c324d_acb7_4615_b70a_8b8b8a044876_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : HasPlanarDrawingOn H)
    (hconn : H.Connected)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ f : ℕ,
      Fintype.card V + f = H.edgeFinset.card + 2 ∧
      3 * f ≤ 2 * H.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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