Finite maximal drawable supergraph

Proved

WilliamsonShmoys.drawable_graph_has_maximal_supergraph__p2m_f3f7a3a7_d237_45a5_b1fd_f31e0f1c8fc4_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph on at least three vertices is contained in a drawable graph on the same vertex set that has no drawable strict edge supergraph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.drawable_graph_has_maximal_supergraph__p2m_f3f7a3a7_d237_45a5_b1fd_f31e0f1c8fc4_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hplanar : HasPlanarDrawingOn K) :
    ∃ J : SimpleGraph V, K.edgeSet ⊆ J.edgeSet ∧
      HasPlanarDrawingOn J ∧
      ∀ L : SimpleGraph V, J.edgeSet ⊂ L.edgeSet → ¬ HasPlanarDrawingOn L := 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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