Finite maximal extension among simple graphs

Proved

exists_maximal_simpleGraph_extension__p2m_87ed5adb_82f3_488c_b808_7f2da9357c21_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, any property of simple graphs that holds of a given graph has a supergraph satisfying the property that is maximal under edge inclusion among graphs with that property.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem exists_maximal_simpleGraph_extension__p2m_87ed5adb_82f3_488c_b808_7f2da9357c21_a1
    {V : Type} [Fintype V]
    (L : SimpleGraph V) (P : SimpleGraph V → Prop) (hP : P L) :
    ∃ H : SimpleGraph V, L ≤ H ∧ P H ∧
      ∀ 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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