Finite simple graphs admit maximal predicate supergraphs
ProvedWilliamsonShmoys.exists_maximal_supergraph_for_predicate__p2m_1e3a6908_3e59_4565_92f0_171f972a39c8_a2approximation-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 maximal among such supergraphs.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_maximal_supergraph_for_predicate__p2m_1e3a6908_3e59_4565_92f0_171f972a39c8_a2
{V : Type} [Fintype V] [LinearOrder V]
(P : SimpleGraph V → Prop)
(H : SimpleGraph V)
(hH : P H) :
∃ K : SimpleGraph V, H ≤ K ∧ P K ∧
∀ K' : SimpleGraph V, K ≤ K' → P K' → K' = 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).