Planar drawing edge-count upper bound
ProvedWilliamsonShmoys.planarDrawingOn_edge_count_upper__p2m_10767c1e_3761_455c_b81d_2c159386481f_a1approximation-algorithmsplanar-graphstreewidth
Every graph with a drawing as defined by HasPlanarDrawingOn, on at least three finite vertices, has at most three times the number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planarDrawingOn_edge_count_upper__p2m_10767c1e_3761_455c_b81d_2c159386481f_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hCard : 3 ≤ Fintype.card V) :
K.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).