Every drawable graph has a maximal drawable supergraph

Proved

WilliamsonShmoys.drawable_graph_has_maximal_drawable_supergraph__p2m_9940682f_4c7e_48c3_97b4_7cba95f9fa9e_a1

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

approximation-algorithmsplanar-graphstreewidth

Any drawable graph on a finite vertex type has a drawable supergraph on the same vertices that has no strictly larger drawable supergraph. Finiteness of the set of possible simple graphs allows choosing a drawable supergraph with maximal edge count. The bound on the number of vertices is not needed here; the edge-count argument is deferred to a separate child.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.drawable_graph_has_maximal_drawable_supergraph__p2m_9940682f_4c7e_48c3_97b4_7cba95f9fa9e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (hK : HasPlanarDrawingOn K) :
    ∃ (L : SimpleGraph V) (hLdec : DecidableRel L.Adj),
      K ≤ L ∧ HasPlanarDrawingOn L ∧
      (∀ M : SimpleGraph V, L < M → ¬ HasPlanarDrawingOn M) := 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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