Connected planar edge bound for at least three vertices
OpenWilliamsonShmoys.connected_planar_edge_bound_large__p2m_2921e68e_aa42_4dea_8127_751b3e371ec8_a1approximation-algorithmsplanar-graphstreewidth
A finite connected simple graph with a crossing-free drawing and at least three vertices has at most three times its vertex count minus three edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.connected_planar_edge_bound_large__p2m_2921e68e_aa42_4dea_8127_751b3e371ec8_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (dG : DecidableRel G.Adj)
(hG : HasPlanarDrawingOn G) (hconn : G.Connected)
(hcard : 3 ≤ Fintype.card V) :
(letI : DecidableRel G.Adj := dG; G.edgeFinset.card) + 3 ≤ 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).