Maximum drawable supergraph exists

Proved

WilliamsonShmoys.planar_drawing_maximal_supergraph__p2m_75eb6681_dffa_41bf_ba37_6fc423d4fd9b_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite graph has a drawable supergraph on the same vertex set whose edge count is at least that of every other drawable supergraph of the original graph.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_maximal_supergraph__p2m_75eb6681_dffa_41bf_ba37_6fc423d4fd9b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V)
    (hMdraw : HasPlanarDrawingOn M) :
    ∃ N : SimpleGraph V, M ≤ N ∧ HasPlanarDrawingOn N ∧
      ∀ N' : SimpleGraph V, M ≤ N' → HasPlanarDrawingOn N' →
        (letI : DecidableRel N'.Adj := Classical.decRel N'.Adj
         N'.edgeFinset.card) ≤
        (letI : DecidableRel N.Adj := Classical.decRel N.Adj
         N.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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