Maximal drawable supergraph extension

Proved

WilliamsonShmoys.planar_supergraph_maximal_extension__p2m_b79a3d22_514c_40c6_b0d9_3192cd4e6368_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 vertex type that is maximal among drawable supergraphs under graph inclusion.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_supergraph_maximal_extension__p2m_b79a3d22_514c_40c6_b0d9_3192cd4e6368_a1 {V : Type} [Fintype V] [LinearOrder V] (K : SimpleGraph V) (dK : DecidableRel K.Adj) (hK : HasPlanarDrawingOn K) : ∃ (L : SimpleGraph V) (dL : DecidableRel L.Adj), K ≤ L ∧ HasPlanarDrawingOn L ∧ ∀ (M : SimpleGraph V) (dM : DecidableRel M.Adj), L ≤ M → HasPlanarDrawingOn M → M ≤ L := 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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