Finite maximal planar supergraph completion

Proved

WilliamsonShmoys.planar_triangulation_maximal_completion__p2m_e3eb1401_8114_44d9_870e_95a93988f448_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite linearly ordered vertex type has a drawable supergraph on the same vertex type that contains all its edges and is maximal under edge inclusion among drawable graphs. The n ≥ 3 hypothesis is retained to match the parent, though finite maximality itself does not need it.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_triangulation_maximal_completion__p2m_e3eb1401_8114_44d9_870e_95a93988f448_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hG : HasPlanarDrawingOn G) :
    ∃ L : SimpleGraph V,
      HasPlanarDrawingOn L ∧
      (∀ u v, G.Adj u v → L.Adj u v) ∧
      (∀ M : SimpleGraph V, HasPlanarDrawingOn M →
        (∀ u v, L.Adj u v → M.Adj u v) →
        ∀ u v, M.Adj u v → L.Adj u v) := 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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