Edge maximal drawable graphs are connected
OpenWilliamsonShmoys.maximal_planar_supergraph_connected__p2m_a4257fd8_446b_43ed_ae2f_f84ba74a352a_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three vertices that has no strictly larger drawable supergraph on the same vertices is connected.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_supergraph_connected__p2m_a4257fd8_446b_43ed_ae2f_f84ba74a352a_a1 {V : Type} [Fintype V] [LinearOrder V] (H : SimpleGraph V) (hH : HasPlanarDrawingOn H) (hcard : 3 ≤ Fintype.card V) (hmax : ∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K → K = H) : H.Connected := 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).