Simple edge bound for a connected planar drawing
ProvedWilliamsonShmoys.planar_drawing_connected_edge_bound__p2m_4939bc54_bf93_4b2e_bbee_bc8378b5882a_a3approximation-algorithmsplanar-graphstreewidth
A connected simple graph with at least three vertices and a crossing-free drawing has at most three times its vertex count minus six edges, counted with Set.ncard to avoid requiring a decidable adjacency instance. A proof route augments the drawing to a triangulation, applies Euler’s formula, and uses the lower bound of three on face boundary lengths. This supplies the connected case used by the parent; formalizing augmentation and the face-count argument remains the core obligation.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_drawing_connected_edge_bound__p2m_4939bc54_bf93_4b2e_bbee_bc8378b5882a_a3
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V)
(hH : HasPlanarDrawingOn H)
(hconn : H.Connected)
(hcard : 3 ≤ Fintype.card V) :
H.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).