Existence of a maximal drawable supergraph

Proved

WilliamsonShmoys.exists_maximal_drawable_supergraph__p2m_28ba4808_169d_4087_9472_de8f0bf86338_a1

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 on the same vertices that is inclusion-maximal among drawable supergraphs and has at least as many edges as the original graph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_drawable_supergraph__p2m_28ba4808_169d_4087_9472_de8f0bf86338_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hH : HasPlanarDrawingOn H) :
    letI : DecidableRel H.Adj := Classical.decRel H.Adj
    ∃ K : SimpleGraph V,
      letI : DecidableRel K.Adj := Classical.decRel K.Adj
      H ≤ K ∧ HasPlanarDrawingOn K ∧
      H.edgeFinset.card ≤ K.edgeFinset.card ∧
      (∀ L : SimpleGraph V,
        letI : DecidableRel L.Adj := Classical.decRel L.Adj
        K ≤ L → HasPlanarDrawingOn L → L ≤ K) := 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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