Existence of a maximal drawable supergraph

Proved

exists_maximal_planar_supergraph__p2m_10eaf2f1_7409_4fae_95fa_e1e70a835601_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite graph on the given vertex set has a drawable supergraph that is maximal under inclusion among drawable supergraphs on that same vertex set.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem exists_maximal_planar_supergraph__p2m_10eaf2f1_7409_4fae_95fa_e1e70a835601_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (dG : DecidableRel G.Adj)
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G) :
    ∃ (M : SimpleGraph V) (dM : DecidableRel M.Adj),
      WilliamsonShmoys.HasPlanarDrawingOn M ∧
      G ≤ M ∧
      ∀ (N : SimpleGraph V),
        M ≤ N → WilliamsonShmoys.HasPlanarDrawingOn N → N ≤ M := 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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