Maximal drawable supergraph extension

Proved

WilliamsonShmoys.planar_drawing_has_maximal_extension__p2m_94636b31_2f9e_4256_93d0_7a9950cdcd71_a1

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

approximation-algorithmsplanar-graphstreewidth

Any drawable graph on a fixed finite ordered vertex set of size at least three has a drawable supergraph on the same vertex set that is maximal among drawable supergraphs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_has_maximal_extension__p2m_94636b31_2f9e_4256_93d0_7a9950cdcd71_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : HasPlanarDrawingOn H)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ K : SimpleGraph V, H ≤ K ∧ HasPlanarDrawingOn K ∧
      ∀ (L : SimpleGraph V) [DecidableRel L.Adj],
        HasPlanarDrawingOn L → K ≤ L → L ≤ K := 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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