Existence of a maximal drawable supergraph

Proved

WilliamsonShmoys.maximal_planar_supergraph__p2m_609d0ef3_f673_413c_a21e_b454ce78643f_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite ordered vertex type is contained in a drawable graph on the same vertices that has no strictly larger drawable supergraph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_supergraph__p2m_609d0ef3_f673_413c_a21e_b454ce78643f_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (W : SimpleGraph V)
    (hW : HasPlanarDrawingOn W) :
    ∃ X : SimpleGraph V, W ≤ X ∧ HasPlanarDrawingOn X ∧
      ∀ Y : SimpleGraph V, X ≤ Y → HasPlanarDrawingOn Y → Y = X := 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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