Existence of a maximal drawable supergraph

Proved

WilliamsonShmoys.maximal_planar_supergraph_exists__p2m_34c1d13a_c463_44f3_a4ff_eb6476f05c93_a2

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on the given finite vertex type has a drawable supergraph that is inclusion-maximal among drawable supergraphs on that same type.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_supergraph_exists__p2m_34c1d13a_c463_44f3_a4ff_eb6476f05c93_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hG : HasPlanarDrawingOn G) :
    ∃ T : SimpleGraph V, G ≤ T ∧ HasPlanarDrawingOn T ∧
      ∀ U : SimpleGraph V, T ≤ U → HasPlanarDrawingOn U → U = T := 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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