Finite maximal supergraph for an arbitrary property

Proved

WilliamsonShmoys.exists_maximal_supergraph_for_property__p2m_b05574a0_b431_4a4f_aba2_4718b07b1171_a1

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

approximation-algorithmsplanar-graphstreewidth

For any property of simple graphs on a finite linearly ordered vertex type, if H satisfies the property then H has a supergraph satisfying it whose edge count is at least that of H and which has no strictly larger feasible supergraph.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_supergraph_for_property__p2m_b05574a0_b431_4a4f_aba2_4718b07b1171_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V → Prop)
    (H : SimpleGraph V)
    (hP : P H) :
    ∃ M : SimpleGraph V,
      H ≤ M ∧
      P M ∧
      (letI : DecidableRel H.Adj := Classical.decRel H.Adj
       letI : DecidableRel M.Adj := Classical.decRel M.Adj
       H.edgeFinset.card ≤ M.edgeFinset.card) ∧
      (∀ N : SimpleGraph V, M ≤ N → P N → N ≤ M) := 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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