Edge count of a maximal planar graph

Proved

WilliamsonShmoys.maximal_planar_edge_count__p2m_cfcb8007_1ff9_4f3a_9061_c47c5589ef6f_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 the same vertex set has exactly three times the number of vertices minus six edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_cfcb8007_1ff9_4f3a_9061_c47c5589ef6f_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hM : HasPlanarDrawingOn M)
    (hmax : ∀ N : SimpleGraph V, HasPlanarDrawingOn N →
      (∀ u v, M.Adj u v → N.Adj u v) →
      ∀ u v, N.Adj u v → M.Adj u v) :
    letI : DecidableRel M.Adj := Classical.decRel M.Adj
    3 * Fintype.card V = M.edgeFinset.card + 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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