Maximal drawable extension on the same vertices
ProvedWilliamsonShmoys.planarDrawing_maximal_extension__p2m_1ec71208_cc1_45fa_ad3a_a9c7012c7f6f_a3approximation-algorithmsplanar-graphstreewidth
A connected drawable graph on at least three vertices has an inclusion-maximal drawable supergraph on the same vertices.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planarDrawing_maximal_extension__p2m_1ec71208_cc1_45fa_ad3a_a9c7012c7f6f_a3
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V)
(hcard : 3 ≤ Fintype.card V)
(hdraw : WilliamsonShmoys.HasPlanarDrawingOn G)
(hconn : G.Connected) :
∃ H : SimpleGraph V,
G ≤ H ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
∀ K : SimpleGraph V, H ≤ K →
WilliamsonShmoys.HasPlanarDrawingOn K → K = H := 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).