Maximal drawable supergraph on a finite vertex type

Proved

WilliamsonShmoys.finite_drawable_supergraph_maximal__p2m_e472cbd3_427a_4623_8f32_b9aff130ed8d_a2

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

approximation-algorithmsplanar-graphstreewidth

For a finite linearly ordered vertex type, every graph with a planar drawing has a drawable supergraph that is maximal among drawable graphs on the same vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.finite_drawable_supergraph_maximal__p2m_e472cbd3_427a_4623_8f32_b9aff130ed8d_a2 {V : Type} [Fintype V] [LinearOrder V] (Q : SimpleGraph V) (hQ : HasPlanarDrawingOn Q) : ∃ R : SimpleGraph V, Q ≤ R ∧ HasPlanarDrawingOn R ∧ ∀ S : SimpleGraph V, R ≤ S → HasPlanarDrawingOn S → S = R := 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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