Maximal drawable graphs are connected
OpenWilliamsonShmoys.maximal_drawable_connected__p2m_6bdbdb3d_755f_4cae_8333_ac9716c72254_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal graph with a crossing-free drawing and at least three vertices is connected. If it were disconnected, an edge could be added between vertices in different components while preserving drawability, contradicting maximality.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_drawable_connected__p2m_6bdbdb3d_755f_4cae_8333_ac9716c72254_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hcard : 3 <= Fintype.card V)
(hmax : forall L : SimpleGraph V, K <= L -> WilliamsonShmoys.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).