Maximal drawing reaches the planar edge bound

Proved

WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_8980508c_eade_47a8_94c5_e0127b6a85f8_a1

by Eyal1990 · Oct 29, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A drawable simple graph on at least three vertices that is inclusion-maximal among drawable supergraphs on the same vertex set has at least 3|V|−6 edges. Maximality forces triangular faces, giving the reverse bound from Euler’s formula and edge-face incidence.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_8980508c_eade_47a8_94c5_e0127b6a85f8_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hV : 3 ≤ Fintype.card V)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hmax : ∀ H' : SimpleGraph V,
      Set.Subset H.edgeSet H'.edgeSet →
      WilliamsonShmoys.HasPlanarDrawingOn H' →
      H'.edgeSet = H.edgeSet) :
    3 * Fintype.card V ≤ H.edgeSet.ncard + 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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