Existence of a maximal planar supergraph

Proved

WilliamsonShmoys.exists_isMaximalPlanarSupergraph__p2m_252969a2_5705_4ea9_b97e_d995b4f5b608_a1

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every graph with a drawing as defined in the target is contained in an inclusion-maximal graph with such a drawing on the same finite vertex type. The finite collection of simple graphs on V makes an inclusion-maximal planar supergraph exist; this does not require the vertex-count lower bound.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_isMaximalPlanarSupergraph__p2m_252969a2_5705_4ea9_b97e_d995b4f5b608_a1 {V : Type} [Fintype V] [LinearOrder V] (L : SimpleGraph V) (hDraw : WilliamsonShmoys.HasPlanarDrawingOn L) : ∃ M : SimpleGraph V, L ≤ M ∧ WilliamsonShmoys.HasPlanarDrawingOn M ∧ ∀ N : SimpleGraph V, M ≤ N → WilliamsonShmoys.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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