Edge bound for connected simple planar graphs
ProvedWilliamsonShmoys.planar_simple_graph_connected_edge_bound__p2m_b11eb619_37a3_43a5_a612_72f4475984ba_a2approximation-algorithmsplanar-graphstreewidth
A finite connected simple graph with a crossing-free drawing and at least three vertices has at most 3|V| − 6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_simple_graph_connected_edge_bound__p2m_b11eb619_37a3_43a5_a612_72f4475984ba_a2 {V : Type} [Fintype V] [LinearOrder V] (G : SimpleGraph V) [DecidableRel G.Adj] (hG : HasPlanarDrawingOn G) (hcard : 3 ≤ Fintype.card V) (hconn : G.Connected) : 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).