Existence of a maximal drawable supergraph

Proved

WilliamsonShmoys_planar_drawing_maximal_extension__p2m_e8e38ca1_7bc9_4d8d_a468_0001acb809e9_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph on the given vertex set has a drawable supergraph that is maximal among drawable supergraphs on that same vertex set. The finite maximality argument establishes this child; the vertex-count hypothesis is retained from the parent.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys_planar_drawing_maximal_extension__p2m_e8e38ca1_7bc9_4d8d_a468_0001acb809e9_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : HasPlanarDrawingOn G)
    (hcard : 3 <= Fintype.card V) :
    ∃ H : SimpleGraph V, G ≤ H ∧ HasPlanarDrawingOn H ∧
      ∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K → K ≤ H := 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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