A drawable graph has a maximal drawable supergraph

Proved

WilliamsonShmoys.planar_maximal_supergraph_extension__p2m_fcfb21bb_7adb_48c4_98c9_78175b7ac091_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph has a drawable supergraph on the same vertices that is maximal among drawable supergraphs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_maximal_supergraph_extension__p2m_fcfb21bb_7adb_48c4_98c9_78175b7ac091_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V)
    (hM : WilliamsonShmoys.HasPlanarDrawingOn M)
    (hcard : 3 <= Fintype.card V) :
    Exists fun N : SimpleGraph V =>
      (forall {u v : V}, M.Adj u v -> N.Adj u v) /\
      WilliamsonShmoys.HasPlanarDrawingOn N /\
      (forall P : SimpleGraph V,
        (forall {u v : V}, N.Adj u v -> P.Adj u v) ->
        WilliamsonShmoys.HasPlanarDrawingOn P -> P = 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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