A finite graph has a maximal supergraph satisfying any witnessed property

Proved

exists_maximal_property_supergraph__p2m_bde7bab7_9900_4674_96b7_2eea7887c7cc_a1

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

approximation-algorithmsplanar-graphstreewidth

For any property of finite simple graphs, if a graph G has that property, then G has a supergraph H with the property that is maximal among the property-satisfying supergraphs of G.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem exists_maximal_property_supergraph__p2m_bde7bab7_9900_4674_96b7_2eea7887c7cc_a1
    {V : Type} [Fintype V]
    (P : SimpleGraph V → Prop)
    (G : SimpleGraph V)
    (hG : P G) :
    ∃ H : SimpleGraph V,
      (∀ u v : V, G.Adj u v → H.Adj u v) ∧
      P H ∧
      (∀ K : SimpleGraph V,
        (∀ u v : V, G.Adj u v → K.Adj u v) →
        P K →
        (∀ u v : V, H.Adj u v → K.Adj u v) →
        K = 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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