Maximal planar drawing implies connectedness
ProvedWilliamsonShmoys.maximal_planar_drawing_connected__p2m_6a6f73b1_aafd_4ec3_b910_d68356c6d6db_a1approximation-algorithmsplanar-graphstreewidth
If a graph on at least three vertices has a crossing-free drawing and has no strictly larger planar supergraph on the same vertex set, then it is connected.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_connected__p2m_6a6f73b1_aafd_4ec3_b910_d68356c6d6db_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
(hmax : ∀ K : SimpleGraph V, G ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K → K = G)
(hcard : 3 ≤ Fintype.card V) :
G.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).