Existence of a maximal drawable supergraph
ProvedWilliamsonShmoys.maximal_drawable_supergraph_exists__p2m_f0debea5_402c_48d1_9751_ec911f78abab_a1approximation-algorithmsplanar-graphstreewidth
Every drawable graph on a finite ordered vertex type has a drawable supergraph on the same vertices that is maximal among drawable supergraphs under edge inclusion.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_drawable_supergraph_exists__p2m_f0debea5_402c_48d1_9751_ec911f78abab_a1 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) [DecidableRel M.Adj] (hM : HasPlanarDrawingOn M) : ∃ N : SimpleGraph V, ∃ (hNdec : DecidableRel N.Adj), M ≤ N ∧ HasPlanarDrawingOn N ∧ (∀ (P : SimpleGraph V) [DecidableRel P.Adj], N ≤ P → HasPlanarDrawingOn P → P = N) := 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).