An inclusion-maximal planar drawing has 3n−6 edges
ProvedWilliamsonShmoys.inclusion_maximal_planar_drawing_has_three_n_minus_six_edges__p2m_6f695848_ebe7_49d1_9027_cfad81c2f288_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal graph admitting the specified crossing-free drawing, on at least three vertices, has exactly 3n−6 edges. The proof must establish that maximality triangulates the faces, then formalize the Euler and face-degree equations for the supplied drawing.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.inclusion_maximal_planar_drawing_has_three_n_minus_six_edges__p2m_6f695848_ebe7_49d1_9027_cfad81c2f288_a1
{V : Type} [Fintype V] [LinearOrder V]
(P : SimpleGraph V) (hP : WilliamsonShmoys.HasPlanarDrawingOn P)
(hmax : ∀ Q : SimpleGraph V, P ≤ Q → WilliamsonShmoys.HasPlanarDrawingOn Q → Q = P)
(hcard : 3 ≤ Fintype.card V) :
letI : DecidableRel P.Adj := Classical.decRel P.Adj
3 * Fintype.card V = P.edgeFinset.card + 6 := 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).