Existence of a maximal drawable extension

Proved

WilliamsonShmoys.exists_maximal_planar_extension__p2m_64019286_52a7_4945_b306_7d1954508408_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite vertex type is contained in a drawable supergraph that is maximal under inclusion among drawable graphs on that same vertex type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_extension__p2m_64019286_52a7_4945_b306_7d1954508408_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H) :
    ∃ K : SimpleGraph V,
      H ≤ K ∧ WilliamsonShmoys.HasPlanarDrawingOn K ∧
        ∀ L : SimpleGraph V, K ≤ L → WilliamsonShmoys.HasPlanarDrawingOn L → L = K := 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).

View graph