Simple planar edge bound from a drawing
OpenWilliamsonShmoys.planarDrawing_edge_count_bound__p2m_75c1dbe5_597b_4ab5_bcaf_dc768f4756d8_a1approximation-algorithmsplanar-graphstreewidth
A finite simple graph with at least three vertices and the supplied crossing-free drawing has at most 3|V|−6 edges. The bound is sharp for a drawn triangle and remains valid for disconnected graphs. One independent route is to apply Euler’s formula componentwise, count edge-side incidences around face boundary walks (counting both sides of bridges), and use that each face boundary has length at least three after handling tree and small-component cases. This removes the planar density estimate from the parent; the remaining work for this child is proving the incidence accounting and treating its boundary cases for the exact drawing definition.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planarDrawing_edge_count_bound__p2m_75c1dbe5_597b_4ab5_bcaf_dc768f4756d8_a1
{V : Type} [Fintype V] [LinearOrder V]
(U : SimpleGraph V) (dU : DecidableRel U.Adj)
(hU : HasPlanarDrawingOn U)
(hcard : 3 ≤ Fintype.card V) :
(letI : DecidableRel U.Adj := dU
U.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).