Every finite planar drawing has a maximal planar supergraph

Proved

WilliamsonShmoys_planar_drawing_maximal_extension__p2m_f305e987_8fa4_48f1_bb88_04ada6b91b35_a2

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

approximation-algorithmsplanar-graphstreewidth

Extend the given planar graph to a planar supergraph on the same finite vertex type that has no strictly larger planar supergraph on that type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys_planar_drawing_maximal_extension__p2m_f305e987_8fa4_48f1_bb88_04ada6b91b35_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hH : WilliamsonShmoys.HasPlanarDrawingOn H) :
    ∃ G : SimpleGraph V, H ≤ G ∧ WilliamsonShmoys.HasPlanarDrawingOn G ∧
      ∀ K : SimpleGraph V, G ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K → K = G := 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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