Edge count of a maximal planar graph

Proved

WilliamsonShmoys.maximal_planar_edge_count__p2m_7b09e600_7df3_4cde_9f3e_028247bb9b4a_a2

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

An edge maximal graph with the supplied planar drawing predicate has 3n − 6 edges when it has at least three vertices.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_7b09e600_7df3_4cde_9f3e_028247bb9b4a_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hplanar : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hcard : 3 ≤ Fintype.card V)
    (hmax : ∀ K : SimpleGraph V, H ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K → K = H) :
    3 * Fintype.card V - 6 = H.edgeSet.ncard := 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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