Finite edge-maximal drawable supergraph
ProvedWilliamsonShmoys.planarDrawingOn_edgeMaximalCompletion__p2m_ff18b25b_a5e2_42d5_a9ba_9dc0688f14d5_a1approximation-algorithmsplanar-graphstreewidth
Every drawable finite graph on a fixed ordered vertex type has a drawable supergraph on the same vertices that is maximal among drawable supergraphs.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planarDrawingOn_edgeMaximalCompletion__p2m_ff18b25b_a5e2_42d5_a9ba_9dc0688f14d5_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) [DecidableRel K.Adj]
(hK : HasPlanarDrawingOn K) :
∃ K' : SimpleGraph V, K ≤ K' ∧ HasPlanarDrawingOn K' ∧
∀ H : SimpleGraph V, K' ≤ H → HasPlanarDrawingOn 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).