Edge count of an edge-maximal drawable graph

Proved

WilliamsonShmoys.maximal_drawable_edge_count__p2m_63ad9a14_da0c_4363_acd0_29ef03d610e6_a2

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

approximation-algorithmsplanar-graphstreewidth

A drawable simple graph on at least three vertices that has no strictly larger drawable supergraph has exactly $3|V|-6$ edges. The remaining work is to derive connectedness and triangular faces from maximality and formalize Euler's formula for the drawing.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_drawable_edge_count__p2m_63ad9a14_da0c_4363_acd0_29ef03d610e6_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (dM : DecidableRel M.Adj)
    (hM : WilliamsonShmoys.HasPlanarDrawingOn M)
    (hn : 3 <= Fintype.card V)
    (hmax : ∀ N : SimpleGraph V,
      WilliamsonShmoys.HasPlanarDrawingOn N →
      M ≤ N →
      N = M) :
    M.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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