Edge count of a maximal planar graph

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WilliamsonShmoys.maximal_planar_edge_count__p2m_a95b2e09_031c_4e18_b347_423c7cdf2425_a1

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

approximation-algorithmsplanar-graphstreewidth

A connected graph on at least three vertices that has a planar drawing and is maximal among drawable supergraphs on the same vertex set has exactly 3|V|−6 edges.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_a95b2e09_031c_4e18_b347_423c7cdf2425_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (J : SimpleGraph V) [DecidableRel J.Adj]
    (hcard : 3 ≤ Fintype.card V)
    (hconn : J.Connected)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn J)
    (hmax : ∀ K : SimpleGraph V, J ≤ K →
      WilliamsonShmoys.HasPlanarDrawingOn K → K = J) :
    J.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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