Every planar drawing has an inclusion-maximal planar supergraph on the same vertices

Proved

WilliamsonShmoys.planar_drawing_has_maximal_supergraph__p2m_86ac92fe_a077_498c_8a86_bf0c24de864c_a1

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

approximation-algorithmsplanar-graphstreewidth

For any finite ordered vertex type, every graph with the specified planar drawing is contained in a planar graph on the same vertices that has no strictly larger planar supergraph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_has_maximal_supergraph__p2m_86ac92fe_a077_498c_8a86_bf0c24de864c_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G) :
    ∃ Q : SimpleGraph V,
      G ≤ Q ∧ WilliamsonShmoys.HasPlanarDrawingOn Q ∧
        ∀ R : SimpleGraph V,
          Q ≤ R → 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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