Finite graph property maximal extension

Proved

WilliamsonShmoys.finite_graph_property_maximal_extension__p2m_e07f1b1a_1ea6_4563_8b26_8f130e22d586_a2

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, if a graph H satisfies a property P and the edge-set cardinalities of all graphs satisfying P have a common finite bound, then H is contained in a P-satisfying graph maximal among P-satisfying supergraphs.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finite_graph_property_maximal_extension__p2m_e07f1b1a_1ea6_4563_8b26_8f130e22d586_a2
    {V : Type} [Fintype V] [DecidableEq V]
    (P : SimpleGraph V → Prop) (H : SimpleGraph V)
    (hH : P H) (B : Nat)
    (hbound : ∀ G : SimpleGraph V, P G → G.edgeSet.ncard ≤ B) :
    ∃ K : SimpleGraph V, H ≤ K ∧ P K ∧
      ∀ L : SimpleGraph V, K ≤ L → P L → L ≤ K := 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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