A drawable extension with maximum edge-set cardinality exists

Proved

WilliamsonShmoys.exists_maximal_drawable_extension_ncard__p2m_f0ae2a89_338a_4b39_88cf_341f6f611f31_a2

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

approximation-algorithmsplanar-graphstreewidth

Among drawable supergraphs of K, there is one with at least as many edges as every other drawable supergraph of K.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_drawable_extension_ncard__p2m_f0ae2a89_338a_4b39_88cf_341f6f611f31_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    ∃ L : SimpleGraph V,
      K ≤ L ∧ WilliamsonShmoys.HasPlanarDrawingOn L ∧
        ∀ M : SimpleGraph V, K ≤ M → WilliamsonShmoys.HasPlanarDrawingOn M →
          M.edgeSet.ncard ≤ L.edgeSet.ncard := 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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