A drawable supergraph maximizing edge count exists

Proved

WilliamsonShmoys.drawable_supergraph_max_cardinality__p2m_6ba26de6_69bc_4fed_ac6d_a483b85550c4_a1

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

approximation-algorithmsplanar-graphstreewidth

Among drawable graphs on the same finite vertex type whose edges include K's edges, there exists one whose edge set has maximum cardinality.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.drawable_supergraph_max_cardinality__p2m_6ba26de6_69bc_4fed_ac6d_a483b85550c4_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (hK : HasPlanarDrawingOn K) :
    Exists fun (L : SimpleGraph V) => And (HasPlanarDrawingOn L)
      (Exists fun (dL : DecidableRel L.Adj) =>
        letI : DecidableRel L.Adj := dL
        And (K.edgeFinset ⊆ L.edgeFinset)
          (forall (M : SimpleGraph V) [DecidableRel M.Adj],
            HasPlanarDrawingOn M -> L.edgeFinset ⊆ M.edgeFinset ->
              M.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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