Planar edge bound for connected graphs
ProvedWilliamsonShmoys.planarEdgeBound_connected__p2m_d0c9e92c_0850_4f5e_91b8_72d4ab522367_a3approximation-algorithmsplanar-graphstreewidth
A connected finite simple graph with a crossing-free drawing and at least three vertices has at most three times its vertex count minus six edges. This isolates the Euler and face-boundary counting argument. A proof route is to derive a plane embedding from the drawing, apply Euler’s formula, and count face-boundary incidences, handling bridges explicitly.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planarEdgeBound_connected__p2m_d0c9e92c_0850_4f5e_91b8_72d4ab522367_a3
{V : Type} [Fintype V] [LinearOrder V]
(L : SimpleGraph V) (dL : DecidableRel L.Adj)
(hL : HasPlanarDrawingOn L)
(hconn : L.Connected)
(hn : 3 ≤ Fintype.card V) :
(letI : DecidableRel L.Adj := dL
L.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).