Face boundary counts from a planar drawing
ProvedWilliamsonShmoys.planar_drawing_face_boundary_count__p2m_e743f516_2fe6_4f90_ae81_e51127ac517d_a3approximation-algorithmsplanar-graphstreewidth
Extract the face boundary lengths of the connected drawing. Every length is at least three, their sum is twice the number of edges, and Euler's relation holds.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
Formal statement
theorem WilliamsonShmoys.planar_drawing_face_boundary_count__p2m_e743f516_2fe6_4f90_ae81_e51127ac517d_a3 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) [DecidableRel M.Adj] (hM : WilliamsonShmoys.HasPlanarDrawingOn M) (hconn : M.Connected) (hcard : Nat.le 3 (Fintype.card V)) : Exists (fun lengths : List Nat => And (forall l : Nat, List.Mem l lengths -> Nat.le 3 l) (And (lengths.sum = 2 * M.edgeFinset.card) (Fintype.card V + lengths.length = M.edgeFinset.card + 2))) := 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).