Existence of an inclusion-maximal drawable supergraph

Proved

WilliamsonShmoys.planar_maximal_supergraph__p2m_2d6ff097_7c5b_40ac_9dcb_907c25524093_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 all drawable supergraphs. Choose a drawable supergraph containing the original graph with the largest edge set; any further drawable supergraph would also contain the original graph, contradicting maximality.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_maximal_supergraph__p2m_2d6ff097_7c5b_40ac_9dcb_907c25524093_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V)
    (hP : WilliamsonShmoys.HasPlanarDrawingOn P) :
    ∃ Q : SimpleGraph V,
      (∀ {u v : V}, P.Adj u v → Q.Adj u v) ∧
      WilliamsonShmoys.HasPlanarDrawingOn Q ∧
      (∀ R : SimpleGraph V,
        (∀ {u v : V}, Q.Adj u v → R.Adj u v) →
        WilliamsonShmoys.HasPlanarDrawingOn R →
        R = Q) := 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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