Euler and face-boundary count for a simple planar drawing

Proved

WilliamsonShmoys.planar_drawing_face_data__p2m_cb41be71_821a_4ee1_8bbc_87748aa8233d_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite connected simple graph with at least three vertices and the supplied crossing-free drawing, there is a natural number f satisfying Euler’s vertex-edge-face equation and the total face-boundary lower bound, expressed as 3f ≤ 2E.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_face_data__p2m_cb41be71_821a_4ee1_8bbc_87748aa8233d_a1 {V : Type} [Fintype V] [LinearOrder V] (K : SimpleGraph V) (dK : DecidableRel K.Adj) (hconn : K.Connected) (hK : WilliamsonShmoys.HasPlanarDrawingOn K) (hcard : 3 <= Fintype.card V) : ∃ f : ℕ, Fintype.card V + f = K.edgeFinset.card + 2 ∧ 3 * f <= 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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