Existence of an inclusion-maximal drawable supergraph

Proved

WilliamsonShmoys.maximal_planar_supergraph__p2m_c8fc71ba_c41f_4610_9114_f9df2f968aaf_a3

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite vertex type has a drawable supergraph on the same vertices that is maximal under adding edges. The proof route is to maximize edge count in the finite family of drawable supergraphs; the child isolates this finite extremal step from the planar edge-count argument.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_supergraph__p2m_c8fc71ba_c41f_4610_9114_f9df2f968aaf_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hN : HasPlanarDrawingOn N) :
    ∃ P : SimpleGraph V,
      (∀ u v, N.Adj u v → P.Adj u v) ∧
      HasPlanarDrawingOn P ∧
      (∀ Q : SimpleGraph V,
        (∀ u v, P.Adj u v → Q.Adj u v) →
        HasPlanarDrawingOn Q →
        (∀ u v, Q.Adj u v → P.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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