Maximal drawable supergraph extension

Proved

WilliamsonShmoys.planar_drawable_maximal_extension__p2m_bba3ebb6_f48a_40f6_8fc3_19f0d2d80e0e_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite ordered vertex type has a drawable supergraph on the same vertices that is maximal under edge inclusion. This follows by maximizing the edge count among the finite set of drawable supergraphs. The condition n ≥ 3 is retained for compatibility with the parent, although this finite maximization step does not need it.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawable_maximal_extension__p2m_bba3ebb6_f48a_40f6_8fc3_19f0d2d80e0e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hG : HasPlanarDrawingOn G) :
    letI : ∀ K : SimpleGraph V, DecidableRel K.Adj :=
      fun K => Classical.decRel K.Adj
    ∃ K : SimpleGraph V,
      G.edgeFinset ⊆ K.edgeFinset ∧
      HasPlanarDrawingOn K ∧
      ∀ L : SimpleGraph V,
        K.edgeFinset ⊆ L.edgeFinset →
        HasPlanarDrawingOn L →
        L.edgeFinset = K.edgeFinset := 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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