Maximal drawable graphs are connected
OpenWilliamsonShmoys.maximal_drawable_graph_connected__p2m_7237be62_bb2d_4386_ba18_ecdd0c420d72_a1approximation-algorithmsplanar-graphstreewidth
If a graph on at least three vertices has a crossing-free drawing and is inclusion-maximal among drawable supergraphs, then it is connected.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_drawable_graph_connected__p2m_7237be62_bb2d_4386_ba18_ecdd0c420d72_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hN : HasPlanarDrawingOn N)
(hmax : ∀ P : SimpleGraph V, N ≤ P → HasPlanarDrawingOn P → P = N) :
N.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).