A finite nonempty set of finite graphs has an inclusion-maximal member
Provedfinite_set_has_maximal_simplegraph_extension__p2m_3dd2f00a_a125_4333_88d9_0f94f3546b41_a2approximation-algorithmsplanar-graphstreewidth
Every finite nonempty set of simple graphs on a finite linearly ordered vertex type contains a graph maximal under inclusion within that set.
Preamble
import Mathlib open SimpleGraph set_option autoImplicit false
Formal statement
theorem finite_set_has_maximal_simplegraph_extension__p2m_3dd2f00a_a125_4333_88d9_0f94f3546b41_a2
{V : Type} [Fintype V] [LinearOrder V]
(S : Set (SimpleGraph V)) (hS : S.Finite) (hne : S.Nonempty) :
∃ K ∈ S, ∀ L ∈ S, K ≤ L → L ≤ K := by sorrySource
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).