Finite maximal drawable extension

Proved

WilliamsonShmoys.drawable_graph_maximal_drawable_extension__p2m_ee01e952_1ee0_4062_a12a_4c7920469f52_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph has a drawable supergraph on the same vertices that is maximal under inclusion among drawable supergraphs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.drawable_graph_maximal_drawable_extension__p2m_ee01e952_1ee0_4062_a12a_4c7920469f52_a1 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) (hG : HasPlanarDrawingOn G) : ∃ M : SimpleGraph V, G ≤ M ∧ HasPlanarDrawingOn M ∧ ∀ N : SimpleGraph V, M ≤ N → HasPlanarDrawingOn N → N = M := 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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