Maximum-edge drawable supergraph exists

Proved

WilliamsonShmoys.maximal_planar_supergraph_by_edge_count__p2m_04c04e16_b235_4c68_889f_e3b37d35b24b_a1

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

approximation-algorithmsplanar-graphstreewidth

Among the drawable supergraphs of a finite drawable graph on the same vertex type, there is one whose edge count is at least that of every other drawable supergraph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_supergraph_by_edge_count__p2m_04c04e16_b235_4c68_889f_e3b37d35b24b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H) :
    ∃ (K : SimpleGraph V),
      (∀ ⦃u v : V⦄, H.Adj u v → K.Adj u v) ∧
      WilliamsonShmoys.HasPlanarDrawingOn K ∧
      (∀ (L : SimpleGraph V),
        (∀ ⦃u v : V⦄, H.Adj u v → L.Adj u v) →
        WilliamsonShmoys.HasPlanarDrawingOn L →
        K.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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