Finite maximal extension for a graph property

Proved

WilliamsonShmoys.finite_property_maximal_extension__p2m_b6765847_4b48_4473_801c_ad92c775fb51_a2

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

approximation-algorithmsplanar-graphstreewidth

For any property of graphs on a finite vertex type, a graph satisfying the property has a supergraph satisfying it that has at least as many edges and admits no larger supergraph with the property.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finite_property_maximal_extension__p2m_b6765847_4b48_4473_801c_ad92c775fb51_a2
    {V : Type} [Fintype V]
    (P : SimpleGraph V → Prop)
    (N : SimpleGraph V)
    (hN : P N) :
    ∃ H : SimpleGraph V,
      N ≤ H ∧
      P H ∧
      N.edgeSet.ncard ≤ H.edgeSet.ncard ∧
      (∀ K : SimpleGraph V,
        H ≤ K → P K → 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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