A drawable extension maximizing edge count

Proved

WilliamsonShmoys.exists_drawable_extension_maximizing_edge_count__p2m_3721485b_3cdc_4a3e_9978_b518aa3e9dbd_a1

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

approximation-algorithmsplanar-graphstreewidth

Among drawable supergraphs of a given drawable graph on a finite ordered vertex type, there is one whose edge set has maximal cardinality.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.exists_drawable_extension_maximizing_edge_count__p2m_3721485b_3cdc_4a3e_9978_b518aa3e9dbd_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hL : HasPlanarDrawingOn L) :
    ∃ T : SimpleGraph V, L ≤ T ∧ HasPlanarDrawingOn T ∧
      ∀ U : SimpleGraph V, L ≤ U → HasPlanarDrawingOn U →
        Nat.card U.edgeSet ≤ Nat.card T.edgeSet := 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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