Finiteness of simple graphs on a finite vertex type
ProvedWilliamsonShmoys.finite_simpleGraph_universe__p2m_15a542b2_5d3b_464a_bec6_5f7385b1f62b_a1approximation-algorithmsplanar-graphstreewidth
The type of simple graphs on a finite vertex type is finite. Encode each graph by its Boolean adjacency matrix; symmetry and irreflexivity only restrict the possible matrices.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.finite_simpleGraph_universe__p2m_15a542b2_5d3b_464a_bec6_5f7385b1f62b_a1
{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).