Maximal planar drawing edge lower bound
ProvedWilliamsonShmoys.maximal_planar_drawing_edge_lower_bound__p2m_b9f43026_a41c_414f_8846_4d67a85cd361_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal graph admitting the specified planar drawing on a fixed vertex type of size at least three has at least three times its number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_edge_lower_bound__p2m_b9f43026_a41c_414f_8846_4d67a85cd361_a1 {V : Type} [Fintype V] [LinearOrder V] (H : SimpleGraph V) (hH : WilliamsonShmoys.HasPlanarDrawingOn H) (hMaximal : forall K : SimpleGraph V, H <= K -> WilliamsonShmoys.HasPlanarDrawingOn K -> K = H) (hCard : 3 <= Fintype.card V) : 3 * Fintype.card V - 6 <= H.edgeSet.ncard := 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).