Edge bound for connected planar drawings
ProvedWilliamsonShmoys.planar_edgeCount_upper_connected__p2m_4d9d61b3_a3d4_4adb_a77f_fb6a2cb8e8dc_a1approximation-algorithmsplanar-graphstreewidth
A connected simple graph with a crossing-free drawing and at least three vertices has at most 3|V|−6 edges. The three-vertex boundary case is included; connected graphs with fewer vertices are outside this lemma's hypotheses.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_edgeCount_upper_connected__p2m_4d9d61b3_a3d4_4adb_a77f_fb6a2cb8e8dc_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) [DecidableRel K.Adj]
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hconn : K.Connected)
(hn : 3 ≤ Fintype.card V) :
K.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).