Existence of a maximal drawable extension

Proved

WilliamsonShmoys.exists_maximal_planar_extension__p2m_af8ffdb6_a3ca_4deb_99d5_c1621ba2f570_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite vertex set has an inclusion-maximal drawable supergraph on that same vertex set. This follows by maximizing edge count over the finite collection of drawable supergraphs; the stated cardinality hypothesis is retained for direct use alongside the edge-count lemma.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_extension__p2m_af8ffdb6_a3ca_4deb_99d5_c1621ba2f570_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hL : HasPlanarDrawingOn L)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ M : SimpleGraph V, L ≤ M ∧ HasPlanarDrawingOn M ∧
      ∀ N : SimpleGraph V, M ≤ N → 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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