An edge-maximal planar graph is connected
OpenWilliamsonShmoys.maximal_planar_graph_connected__p2m_801cf2e4_6b6e_4272_a43f_109c3b8f7482_a2approximation-algorithmsplanar-graphstreewidth
A finite graph with a crossing-free drawing that is maximal under edge inclusion among planar supergraphs on the same vertices is connected.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_graph_connected__p2m_801cf2e4_6b6e_4272_a43f_109c3b8f7482_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hK : HasPlanarDrawingOn K)
(hmax : ∀ L : SimpleGraph V,
(∀ u v, K.Adj u v → L.Adj u v) →
HasPlanarDrawingOn L →
∀ u v, L.Adj u v → K.Adj u v) :
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).