An inclusion-maximal drawable graph is connected
OpenWilliamsonShmoys.inclusion_maximal_planar_drawing_is_connected__p2m_47afe67c_60e1_47f9_a634_1f5964e2e86e_a3approximation-algorithmsplanar-graphstreewidth
A drawable graph that is maximal among all drawable supergraphs on the same finite vertex type is connected.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.inclusion_maximal_planar_drawing_is_connected__p2m_47afe67c_60e1_47f9_a634_1f5964e2e86e_a3
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V)
(hM : WilliamsonShmoys.HasPlanarDrawingOn M)
(hmax : ∀ N : SimpleGraph V, M ≤ N → WilliamsonShmoys.HasPlanarDrawingOn N → N = M) :
M.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).