Maximal planar drawing has the Euler edge count

Proved

WilliamsonShmoys.maximal_planar_drawing_has_euler_edge_count__p2m_07094302_93b5_40e3_90fd_090a7e1ed5c9_a1

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

approximation-algorithmsplanar-graphstreewidth

A drawable simple graph on at least three vertices that is maximal among drawable supergraphs on its vertex set has exactly three times its vertex count minus six edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_has_euler_edge_count__p2m_07094302_93b5_40e3_90fd_090a7e1ed5c9_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (dH : DecidableRel H.Adj)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hn : 3 <= Fintype.card V)
    (hmax : ∀ (J : SimpleGraph V), H ≤ J →
      WilliamsonShmoys.HasPlanarDrawingOn J → J ≤ H) :
    (letI : DecidableRel H.Adj := dH;
      H.edgeFinset.card = 3 * Fintype.card V - 6) := 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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