Lower bound for a maximal drawable graph

Proved

WilliamsonShmoys.maximal_planar_edge_count_lower_bound__p2m_64c9133b_a623_45e5_bf52_4877372e6864_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 graphs on the same vertex set has at least 3|V|−6 edges. Extend its drawing to a triangulation on that vertex set; maximality forces the extension to have no additional edges, and Euler’s formula gives the bound.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count_lower_bound__p2m_64c9133b_a623_45e5_bf52_4877372e6864_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V)
    (hP : WilliamsonShmoys.HasPlanarDrawingOn P)
    (hcard : 3 <= Fintype.card V)
    (hmax : ∀ Q : SimpleGraph V,
      (∀ {u v : V}, P.Adj u v → Q.Adj u v) →
      WilliamsonShmoys.HasPlanarDrawingOn Q →
      (∀ {u v : V}, Q.Adj u v → P.Adj u v)) :
    letI : DecidableRel P.Adj := Classical.decRel P.Adj
    3 * Fintype.card V - 6 <= P.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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