Finite maximal planar extension

Proved

WilliamsonShmoys.maximal_planar_extension__p2m_63f4596f_16e9_46f5_8d7e_b4c7e853b2c9_a1__p2m_63f4596f_16e9_46f5_8d7e_b4c7e853b2c9_a2

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite ordered vertex type is contained in a drawable graph that is maximal under adding edges on that same vertex type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_extension__p2m_63f4596f_16e9_46f5_8d7e_b4c7e853b2c9_a1__p2m_63f4596f_16e9_46f5_8d7e_b4c7e853b2c9_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V) (hP : HasPlanarDrawingOn P) :
    ∃ H : SimpleGraph V,
      P ≤ H ∧ HasPlanarDrawingOn H ∧
      (∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K → K = H) := 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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