Edge count of a maximal planar drawing

Proved

WilliamsonShmoys.maximal_planar_drawing_has_euler_edge_count__p2m_64f9a932_9abc_4892_8345_6d855cf8c7f4_a1

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

approximation-algorithmsplanar-graphstreewidth

For a graph on at least three finite linearly ordered vertices, if it has the supplied planar drawing and no drawable supergraph on those vertices properly extends it, then its edge count is exactly three times the number of vertices minus six.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_has_euler_edge_count__p2m_64f9a932_9abc_4892_8345_6d855cf8c7f4_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hV : 3 <= Fintype.card V)
    (hK : HasPlanarDrawingOn K)
    (hmax : ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L ≤ K) :
    (letI : DecidableRel K.Adj := Classical.decRel K.Adj;
     3 * Fintype.card V - 6 = 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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