Finite maximal qualifying supergraph

Proved

WilliamsonShmoys.finite_maximal_qualifying_supergraph__p2m_7b028f0e_9557_4d63_b320_a3debd61f7f1_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For any property P holding of G, there is a qualifying supergraph of G that has no strict qualifying supergraph above it.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finite_maximal_qualifying_supergraph__p2m_7b028f0e_9557_4d63_b320_a3debd61f7f1_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (P : SimpleGraph V → Prop)
    (hP : P G) :
    ∃ H : SimpleGraph V, G ≤ H ∧ P H ∧
      ∀ H' : SimpleGraph V, G ≤ H' → P H' → ¬ H < H' := 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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