Maximal drawable supergraph on the same vertices

Proved

WilliamsonShmoys.drawable_graph_has_maximal_planar_supergraph__p2m_9a3366a6_4a3c_4f33_8539_99f470118709_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite linearly ordered vertex type has a drawable supergraph on the same vertices that is maximal among drawable supergraphs, when there are at least three vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.drawable_graph_has_maximal_planar_supergraph__p2m_9a3366a6_4a3c_4f33_8539_99f470118709_a1 {V : Type} [Fintype V] [LinearOrder V] (L : SimpleGraph V) (dL : DecidableRel L.Adj) (hL : HasPlanarDrawingOn L) (hcard : 3 <= Fintype.card V) : ∃ (M : SimpleGraph V) (dM : DecidableRel M.Adj), L <= M ∧ HasPlanarDrawingOn M ∧ ∀ (N : SimpleGraph V) (dN : DecidableRel N.Adj), 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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