Euler and face-incidence counts for maximal planar drawings

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WilliamsonShmoys.maximalPlanarDrawingOn_euler_face_counts__p2m_77e6a9aa_019b_4811_bd18_07d7e3258778_a1

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

approximation-algorithmsplanar-graphstreewidth

For a drawable graph on at least three finite ordered vertices that is maximal among drawable supergraphs on the same vertex type, there is a natural number f satisfying the triangular face-incidence count and Euler's formula.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximalPlanarDrawingOn_euler_face_counts__p2m_77e6a9aa_019b_4811_bd18_07d7e3258778_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hCard : 3 ≤ Fintype.card V)
    (hmax : ∀ L : SimpleGraph V, K ≤ L → WilliamsonShmoys.HasPlanarDrawingOn L → L ≤ K) :
    ∃ f : ℕ,
      2 * K.edgeSet.ncard = 3 * f ∧
      Fintype.card V + f = K.edgeSet.ncard + 2 := 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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