A maximal planar supergraph exists

Proved

WilliamsonShmoys.maximal_planar_supergraph__p2m_8e16201b_1194_4bde_8c0d_f203800a8dc6_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite drawn graph has a planar supergraph that contains its edges, has at least as many edges, and is maximal under planar supergraph inclusion. The finite search over simple graphs on V, maximizing edge count among planar supergraphs of K, is a direct proof route. This removes the parent’s finite maximality and edge-count bookkeeping; the parent still needs the topological strict-extension lemma to show this maximal graph is connected.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_supergraph__p2m_8e16201b_1194_4bde_8c0d_f203800a8dc6_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (dK : DecidableRel K.Adj)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    Exists fun (H : SimpleGraph V) =>
      Exists fun (dH : DecidableRel H.Adj) =>
        And (K ≤ H)
          (And (WilliamsonShmoys.HasPlanarDrawingOn H)
            (And (Nat.le K.edgeFinset.card H.edgeFinset.card)
              (∀ (J : SimpleGraph V), ∀ (dJ : DecidableRel J.Adj),
                K ≤ J → WilliamsonShmoys.HasPlanarDrawingOn J →
                  H ≤ 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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