Maximal extension for a finite hereditary graph property

Proved

finite_maximal_extension_of_hereditary_property__p2m_3e187282_ed42_4b71_bc6a_a0a726731916_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, any graph satisfying a downward-closed property has an extension that is maximal among graphs satisfying that property.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem finite_maximal_extension_of_hereditary_property__p2m_3e187282_ed42_4b71_bc6a_a0a726731916_a1
    {V : Type} [Fintype V] [DecidableEq V]
    (P : SimpleGraph V → Prop)
    (hhered : ∀ ⦃A B : SimpleGraph V⦄, A ≤ B → P B → P A)
    (G : SimpleGraph V) (hG : P G) :
    ∃ R : SimpleGraph V, G ≤ R ∧ P R ∧
      ∀ H : SimpleGraph V, R ≤ H → P H → H = R := 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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