Simple planar drawing edge upper bound
ProvedWilliamsonShmoys.planar_drawing_edge_upper_bound__p2m_b9f43026_a41c_414f_8846_4d67a85cd361_a1approximation-algorithmsplanar-graphstreewidth
Every simple graph on at least three vertices with the specified crossing-free drawing has at most 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.planar_drawing_edge_upper_bound__p2m_b9f43026_a41c_414f_8846_4d67a85cd361_a1 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G) (hCard : 3 <= Fintype.card V) : G.edgeSet.ncard <= 3 * Fintype.card V - 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).