Maximal planar graph is connected
OpenWilliamsonShmoys.maximal_planar_connected__p2m_b2f31750_6395_4f01_b4dd_f5215a908d6d_a1approximation-algorithmsplanar-graphstreewidth
A maximal planar graph with at least three vertices is connected. This isolates the step needed to apply the connected form of Euler’s formula. A proof route is to use a planar drawing to connect two distinct components by an arc in a complementary region, then contradict maximality. The vertex lower bound excludes degenerate empty and singleton cases.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_connected__p2m_b2f31750_6395_4f01_b4dd_f5215a908d6d_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V)
(hN : WilliamsonShmoys.HasPlanarDrawingOn N)
(hcard : 3 ≤ Fintype.card V)
(hmax : ∀ P : SimpleGraph V, N ≤ P → WilliamsonShmoys.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).