Extract Euler and face-incidence data from a planar drawing
DisprovedWilliamsonShmoys.drawing_to_euler_face_certificate__p2m_e2583d82_96a1_4ed4_8709_65222159201c_a4approximation-algorithmsplanar-graphstreewidth
A connected graph with the given crossing-free arc drawing has a natural face count satisfying Euler’s equation and the face-incidence bound.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.drawing_to_euler_face_certificate__p2m_e2583d82_96a1_4ed4_8709_65222159201c_a4
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hconn : K.Connected)
(hK : HasPlanarDrawingOn K) :
letI : (L : SimpleGraph V) -> DecidableRel L.Adj :=
fun L => Classical.decRel L.Adj
∃ f : ℕ,
Fintype.card V + f = K.edgeFinset.card + 2 ∧
3 * f ≤ 2 * K.edgeFinset.card := 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).