Euler and face-incidence equations for a maximal planar drawing

Proved

WilliamsonShmoys.maximal_planar_face_decomposition__p2m_f5f2a8b8_a8d9_45a6_8446_39002bc3ddba_a8

by Eyal1990 · Oct 29, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal drawable graph on at least three vertices has a face count satisfying Euler's equation and the triangular face-edge incidence equation.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_face_decomposition__p2m_f5f2a8b8_a8d9_45a6_8446_39002bc3ddba_a8
    {V : Type} [Fintype V] [LinearOrder V]
    (K H : SimpleGraph V)
    (hV : LE.le 3 (Fintype.card V))
    (hsub : Set.Subset K.edgeSet H.edgeSet)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hmax : ∀ H' : SimpleGraph V,
      Set.Subset H.edgeSet H'.edgeSet →
      WilliamsonShmoys.HasPlanarDrawingOn H' →
      H'.edgeSet = H.edgeSet) :
    ∃ f : ℕ,
      Fintype.card V + f = H.edgeSet.ncard + 2 ∧
      3 * f = 2 * H.edgeSet.ncard := 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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