Finite maximal supergraph for an arbitrary property
ProvedWilliamsonShmoys.exists_maximal_supergraph_for_property__p2m_d747e68f_4b38_43ce_8028_cff46ecd3e2f_a2approximation-algorithmsplanar-graphstreewidth
For a finite vertex type, every graph satisfying a property has a supergraph satisfying that property which is maximal among its supergraphs.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_maximal_supergraph_for_property__p2m_d747e68f_4b38_43ce_8028_cff46ecd3e2f_a2 {V : Type} [Fintype V] (L : SimpleGraph V) (P : SimpleGraph V → Prop) (hP : P L) : ∃ M : SimpleGraph V, L ≤ M ∧ P M ∧ ∀ N : SimpleGraph V, M ≤ N → P N → N = M := 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).