The simple planar edge bound
ProvedWilliamsonShmoys_planar_edge_bound_from_drawing__p2m_78e6a567_9e35_4430_95ce_28355556e16a_a2approximation-algorithmsplanar-graphstreewidth
A connected simple planar graph on at least three vertices has at most 3n − 6 edges. One proof route extends the drawing to a maximal simple plane graph on the same vertices, whose faces are triangular. Combined with the Euler child, this removes the requested face-boundary inequality from the parent proof.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib
Formal statement
theorem WilliamsonShmoys_planar_edge_bound_from_drawing__p2m_78e6a567_9e35_4430_95ce_28355556e16a_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hconn : K.Connected)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
K.edgeSet.ncard ≤ 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).