Existence of an edge-maximal planar supergraph

Proved

WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_801cf2e4_6b6e_4272_a43f_109c3b8f7482_a2

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

approximation-algorithmsplanar-graphstreewidth

Given a finite graph with a crossing-free drawing, there is a planar supergraph on the same vertices that is maximal under edge inclusion among planar graphs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_801cf2e4_6b6e_4272_a43f_109c3b8f7482_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hH : HasPlanarDrawingOn H) :
    ∃ K : SimpleGraph V,
      (∀ u v, H.Adj u v → K.Adj u v) ∧
      HasPlanarDrawingOn K ∧
      ∀ L : SimpleGraph V,
        (∀ u v, K.Adj u v → L.Adj u v) →
        HasPlanarDrawingOn L →
        ∀ u v, L.Adj u v → K.Adj u v := 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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