Maximal supergraph satisfying a property on a finite vertex type
ProvedWilliamsonShmoys.exists_maximal_supergraph_for_property__p2m_62c66bf3_533d_43c1_bde3_37ac00c5d3e8_a2approximation-algorithmsplanar-graphstreewidth
On a finite vertex type, any graph property satisfied by a graph L is satisfied by some supergraph of L that has no strictly larger supergraph satisfying the property.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_maximal_supergraph_for_property__p2m_62c66bf3_533d_43c1_bde3_37ac00c5d3e8_a2
{V : Type} [Fintype V] [LinearOrder V]
(L : SimpleGraph V) (P : SimpleGraph V → Prop)
(hL : P L) :
∃ K : SimpleGraph V, L ≤ K ∧ P K ∧
(∀ H : SimpleGraph V, K ≤ H → P H → H ≤ 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).