An inclusion-maximal drawable supergraph exists

Proved

WilliamsonShmoys.exists_inclusion_maximal_planar_supergraph__p2m_6f695848_ebe7_49d1_9027_cfad81c2f288_a1

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

approximation-algorithmsplanar-graphstreewidth

Among the drawable supergraphs of N on the same finite vertex type, there is one that has no strictly larger drawable supergraph. The proof is finite maximization over edge sets; n≥3 is retained to match the parent and the edge-count result.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_inclusion_maximal_planar_supergraph__p2m_6f695848_ebe7_49d1_9027_cfad81c2f288_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (hN : WilliamsonShmoys.HasPlanarDrawingOn N)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ P : SimpleGraph V, N ≤ P ∧ WilliamsonShmoys.HasPlanarDrawingOn P ∧
      ∀ Q : SimpleGraph V, P ≤ Q → WilliamsonShmoys.HasPlanarDrawingOn Q → Q = P := 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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