Maximal predicate-satisfying supergraph on a finite vertex type
ProvedWilliamsonShmoys.exists_maximal_supergraph_of_finite__p2m_1bbb274f_042a_4c5e_8e43_fd05505ca5a4_a1approximation-algorithmsplanar-graphstreewidth
For any predicate on simple graphs over a finite linearly ordered vertex type, if H satisfies the predicate, then H has a predicate-satisfying supergraph that is maximal under edge inclusion among predicate-satisfying supergraphs.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_maximal_supergraph_of_finite__p2m_1bbb274f_042a_4c5e_8e43_fd05505ca5a4_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (P : SimpleGraph V → Prop) (hP : P H) :
∃ K : SimpleGraph V, P K ∧
(∀ u v, H.Adj u v → K.Adj u v) ∧
(∀ L : SimpleGraph V, P L →
(∀ u v, K.Adj u v → L.Adj u v) →
(∀ u v, L.Adj u v → K.Adj u v)) := 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).