Euler upper bound for drawable simple graphs
ProvedWilliamsonShmoys.planar_drawing_edge_count_upper_bound__p2m_e087affc_3721_47e3_9583_19a35624e79d_a1approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on at least three vertices has at most three times its vertex count minus six edges. This includes disconnected graphs.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_edge_count_upper_bound__p2m_e087affc_3721_47e3_9583_19a35624e79d_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
(hcard : 3 ≤ Fintype.card V) :
letI : DecidableRel G.Adj := Classical.decRel G.Adj
G.edgeFinset.card + 6 ≤ 3 * Fintype.card V := 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).