Finite drawable edge maximizer

Proved

WilliamsonShmoys.finite_drawable_edge_maximizer__p2m_0117d7a7_55cc_4763_b339_05a114946f12_a2

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite ordered vertex type has a drawable supergraph whose edge set has maximum cardinality among drawable supergraphs of the original graph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finite_drawable_edge_maximizer__p2m_0117d7a7_55cc_4763_b339_05a114946f12_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V)
    (hL : WilliamsonShmoys.HasPlanarDrawingOn L) :
    ∃ M : SimpleGraph V,
      L ≤ M ∧ WilliamsonShmoys.HasPlanarDrawingOn M ∧
        ∀ N : SimpleGraph V,
          M ≤ N → WilliamsonShmoys.HasPlanarDrawingOn N →
            N.edgeSet.ncard ≤ M.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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