Existence of an edge-maximal drawable supergraph

Proved

WilliamsonShmoys.maximal_drawable_supergraph_exists__p2m_63ad9a14_da0c_4363_acd0_29ef03d610e6_a2

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 graph inclusion. Since there are finitely many simple graphs on the vertex type, maximize edge count among drawable supergraphs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_drawable_supergraph_exists__p2m_63ad9a14_da0c_4363_acd0_29ef03d610e6_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V)
    (hL : WilliamsonShmoys.HasPlanarDrawingOn L) :
    ∃ (M : SimpleGraph V) (dM : DecidableRel M.Adj),
      WilliamsonShmoys.HasPlanarDrawingOn M ∧
      L ≤ M ∧
      (∀ N : SimpleGraph V,
        WilliamsonShmoys.HasPlanarDrawingOn N →
        M ≤ N →
        N = 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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