Every drawable finite graph has an inclusion-maximal drawable supergraph

Proved

WilliamsonShmoys.exists_inclusion_maximal_planar_supergraph__p2m_261600ee_7fa4_4a45_a37b_002442c9d475_a2

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

approximation-algorithmsplanar-graphstreewidth

On a fixed finite vertex set, choose a drawable graph containing G with the largest possible edge set. Finiteness of the candidate graph family gives an inclusion-maximal choice. This removes the extension-existence step from the parent; a proof can encode candidate edge sets as a finite family and take a maximum. No vertex-count lower bound is needed here.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.exists_inclusion_maximal_planar_supergraph__p2m_261600ee_7fa4_4a45_a37b_002442c9d475_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G) :
    ∃ M : SimpleGraph V,
      G ≤ M ∧ WilliamsonShmoys.HasPlanarDrawingOn 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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