Existence of a maximal drawable supergraph

Proved

WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_4b25ff07_58d5_4fd9_86ca_512a78018f09_a2

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

approximation-algorithmsplanar-graphstreewidth

Every finite drawable graph has a drawable supergraph on the same vertex set that is maximal under edge inclusion among drawable supergraphs. Prove this by maximizing edge count among the finite family of such graphs; if a drawable supergraph strictly contained more edges, it would contradict maximality of the count.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_supergraph__p2m_4b25ff07_58d5_4fd9_86ca_512a78018f09_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hK : HasPlanarDrawingOn K) :
    letI : ∀ L : SimpleGraph V, DecidableRel L.Adj :=
      fun L => Classical.decRel L.Adj
    ∃ L : SimpleGraph V,
      K.edgeFinset ⊆ L.edgeFinset ∧
      HasPlanarDrawingOn L ∧
      ∀ M : SimpleGraph V,
        L.edgeFinset ⊆ M.edgeFinset →
        HasPlanarDrawingOn M →
        M.edgeFinset = L.edgeFinset := 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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