A planar drawing has a maximal same-vertex-set planar supergraph

Proved

WilliamsonShmoys.planar_drawing_has_maximal_planar_supergraph__p2m_80e34e39_9312_4353_9ec3_c3ff17c125fa_a1

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Any graph with the supplied crossing-free drawing has a supergraph on the same vertices that admits such a drawing and is maximal under edge inclusion among graphs admitting such a drawing.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

Formal statement

theorem WilliamsonShmoys.planar_drawing_has_maximal_planar_supergraph__p2m_80e34e39_9312_4353_9ec3_c3ff17c125fa_a1 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) (hDraw : WilliamsonShmoys.HasPlanarDrawingOn G) : ∃ H : SimpleGraph V, G.edgeSet ⊆ H.edgeSet ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧ (∀ K : SimpleGraph V, H.edgeSet ⊆ K.edgeSet → WilliamsonShmoys.HasPlanarDrawingOn K → K.edgeSet = H.edgeSet) := 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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