Extend a drawable graph to a maximal drawable graph

Proved

WilliamsonShmoys.planarDrawingOn_maximalExtension__p2m_bc222e64_5fd0_4544_bad9_127a7d6f7d68_a2

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

approximation-algorithmsplanar-graphstreewidth

A graph drawable on a finite ordered vertex type has a maximal drawable supergraph on that same vertex type. Choose a drawable supergraph with the largest possible finite edge set; any drawable supergraph of it must have the same edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planarDrawingOn_maximalExtension__p2m_bc222e64_5fd0_4544_bad9_127a7d6f7d68_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hH : HasPlanarDrawingOn H) :
    ∃ K : SimpleGraph V, H ≤ K ∧ HasPlanarDrawingOn K ∧
      ∀ K' : SimpleGraph V, K ≤ K' → HasPlanarDrawingOn K' → K' = 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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