Edge count of a maximal drawable graph

Proved

WilliamsonShmoys.edge_count_of_maximal_planar_drawing__p2m_b6ef5ab1_c465_4724_83b5_1fd8c793e204_a1

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

approximation-algorithmsplanar-graphstreewidth

A drawable graph on a finite linearly ordered vertex type with at least three vertices, maximal among drawable supergraphs on that type, has exactly three times the number of vertices minus six edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.edge_count_of_maximal_planar_drawing__p2m_b6ef5ab1_c465_4724_83b5_1fd8c793e204_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 <= Fintype.card V)
    (hK : HasPlanarDrawingOn K)
    (hmax : forall L : SimpleGraph V, LE.le K L -> HasPlanarDrawingOn L -> Eq L K) :
    letI : (G : SimpleGraph V) -> DecidableRel G.Adj := fun G => Classical.decRel G.Adj
    Eq K.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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