Existence of a maximal drawable supergraph

Proved

WilliamsonShmoys.exists_maximal_drawable_supergraph__p2m_c985394f_ff9a_4827_826c_d7a45e5d518a_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 vertex set that is maximal among drawable supergraphs under edge inclusion. The proof route is finite maximal-element selection, since there are finitely many simple graphs on the fixed vertex set and the original graph is an eligible member.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_drawable_supergraph__p2m_c985394f_ff9a_4827_826c_d7a45e5d518a_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hdraw : HasPlanarDrawingOn G) :
    letI : forall H : SimpleGraph V, DecidableRel H.Adj := fun H => Classical.decRel H.Adj
    exists N : SimpleGraph V,
      (forall u v, G.Adj u v -> N.Adj u v) /\
      HasPlanarDrawingOn N /\
      forall H : SimpleGraph V,
        (forall u v, N.Adj u v -> H.Adj u v) ->
        HasPlanarDrawingOn H -> H = 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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