A drawable graph has a maximum-edge drawable supergraph

Proved

WilliamsonShmoys.exists_maximum_planar_supergraph__p2m_c11f1659_d0fc_4751_a84f_f498781343a5_a1

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

approximation-algorithmsplanar-graphstreewidth

Among the finitely many drawable simple graphs on the given finite vertex type, there is one with maximum edge count that contains the given drawable graph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximum_planar_supergraph__p2m_c11f1659_d0fc_4751_a84f_f498781343a5_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (dH : DecidableRel H.Adj)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H) :
    ∃ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
      WilliamsonShmoys.HasPlanarDrawingOn K ∧
      H.edgeFinset ⊆ K.edgeFinset ∧
      ∀ (L : SimpleGraph V) (dL : DecidableRel L.Adj),
        WilliamsonShmoys.HasPlanarDrawingOn L →
        K.edgeFinset.card ≥ L.edgeFinset.card := 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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