Maximal drawable graph is connected
OpenWilliamsonShmoys.maximal_planar_drawing_connected__p2m_c14927fd_3bb2_4aa2_8e62_690259e960a9_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three vertices that is maximal among drawable supergraphs on the same vertices is connected. This isolates the geometric argument needed by the parent: if the graph were disconnected, its drawing could be extended by an edge between components.
Preamble
import Mathlib import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_connected__p2m_c14927fd_3bb2_4aa2_8e62_690259e960a9_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hK : HasPlanarDrawingOn K)
(hmax : ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L ≤ K) :
K.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).