Maximal drawable supergraph extension

Proved

drawable_graph_has_maximal_drawable_extension__p2m_8b622a66_c72a_42ac_86d2_b24f5cfdd6f7_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite ordered vertex type has an inclusion-maximal drawable supergraph on the same vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem drawable_graph_has_maximal_drawable_extension__p2m_8b622a66_c72a_42ac_86d2_b24f5cfdd6f7_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H) :
    ∃ K : SimpleGraph V,
      WilliamsonShmoys.HasPlanarDrawingOn K ∧
      H ≤ K ∧
      ∀ L : SimpleGraph V,
        WilliamsonShmoys.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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