Edge bound for a connected planar graph
OpenWilliamsonShmoys.connected_planar_edge_bound__p2m_a60c0160_4029_4185_bfe8_ff4b14f19470_a2approximation-algorithmsplanar-graphstreewidth
A connected finite simple graph with at least three vertices and the specified crossing-free drawing has at most 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.connected_planar_edge_bound__p2m_a60c0160_4029_4185_bfe8_ff4b14f19470_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) [DecidableRel H.Adj]
(hconn : H.Connected)
(hdraw : WilliamsonShmoys.HasPlanarDrawingOn H)
(hn : 3 ≤ Fintype.card V) :
H.edgeFinset.card ≤ 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).