Finiteness of simple graphs on a finite vertex type
ProvedWilliamsonShmoys.finite_simple_graph_type__p2m_7421bb86_5406_4559_80b0_0937bd98493b_a2approximation-algorithmsplanar-graphstreewidth
The type of simple graphs on a finite vertex type is finite. This supplies the finiteness hypothesis needed to apply the abstract maximal-extension result to graphs. A proof route encodes each graph by its set of edges, a subset of the finite type of unordered vertex pairs; this also covers an empty vertex type.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.finite_simple_graph_type__p2m_7421bb86_5406_4559_80b0_0937bd98493b_a2 {V : Type} [Fintype V] : Finite (SimpleGraph 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).