Finite maximal planar extension

Proved

WilliamsonShmoys.planar_extension_to_edge_maximal__p2m_9aae8c1a_0088_4039_a6d9_b99fa0bd700a_a1

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 on the same vertices that has no strictly larger drawable supergraph on those vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_extension_to_edge_maximal__p2m_9aae8c1a_0088_4039_a6d9_b99fa0bd700a_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V)
    (hM : WilliamsonShmoys.HasPlanarDrawingOn M) :
    ∃ N : SimpleGraph V,
      M ≤ N ∧ WilliamsonShmoys.HasPlanarDrawingOn N ∧
        ∀ K : SimpleGraph V,
          N ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K → K = N := 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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