Finite maximal extension for a hereditary graph property
ProvedWilliamsonShmoys.exists_maximal_good_supergraph__p2m_3bb112e2_2df9_4e3a_a157_84b2375941f4_a2approximation-algorithmsplanar-graphstreewidth
On a finite vertex type, every graph satisfying a property closed under subgraphs is contained in a supergraph maximal among graphs with that property.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_maximal_good_supergraph__p2m_3bb112e2_2df9_4e3a_a157_84b2375941f4_a2
{V : Type} [Fintype V]
(P : SimpleGraph V) (Good : SimpleGraph V → Prop)
(hP : Good P)
(hsub : ∀ {G H : SimpleGraph V}, H ≤ G → Good G → Good H) :
∃ Q : SimpleGraph V, P ≤ Q ∧ Good Q ∧
∀ R : SimpleGraph V, Q ≤ R → Good R → R ≤ Q := 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).