Finite maximal drawable supergraph

Proved

WilliamsonShmoys.planar_drawable_maximal_supergraph__p2m_e31b05a7_4e78_4789_8a19_ecdeedbcdfa7_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 a drawable supergraph on the same vertices that is maximal among drawable supergraphs under edge inclusion. This removes the finite graph-choice and maximality bookkeeping from the parent. The child follows by selecting a maximal element from the finite nonempty set of drawable supergraphs of M; the chosen graph is drawable and M itself witnesses nonemptiness. No cardinality lower bound is needed here.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawable_maximal_supergraph__p2m_e31b05a7_4e78_4789_8a19_ecdeedbcdfa7_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V)
    (hM : WilliamsonShmoys.HasPlanarDrawingOn M) :
    ∃ N : SimpleGraph V,
      M ≤ N ∧ WilliamsonShmoys.HasPlanarDrawingOn N ∧
        ∀ P : SimpleGraph V, N ≤ P → WilliamsonShmoys.HasPlanarDrawingOn P → P = N := 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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