Extend a planar drawing to a maximal planar supergraph

Proved

WilliamsonShmoys.planar_drawing_has_maximal_planar_supergraph__p2m_07094302_93b5_40e3_90fd_090a7e1ed5c9_a1

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

approximation-algorithmsplanar-graphstreewidth

Any drawable finite simple graph has a drawable supergraph on the same vertices that is maximal among drawable supergraphs under edge inclusion.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_has_maximal_planar_supergraph__p2m_07094302_93b5_40e3_90fd_090a7e1ed5c9_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    ∃ (H : SimpleGraph V) (dH : DecidableRel H.Adj),
      K ≤ H ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
        ∀ (J : SimpleGraph V), H ≤ J →
          WilliamsonShmoys.HasPlanarDrawingOn J → J ≤ H := 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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