Face-count certificate from a connected planar drawing
OpenWilliamsonShmoys.connected_planar_drawing_face_count_certificate__p2m_230ef2ff_d0ad_4b7a_9595_07bea2af44fc_a1approximation-algorithmsplanar-graphstreewidth
A crossing-free drawing of a finite connected simple graph with at least three vertices induces a finite set of faces whose count satisfies Euler’s formula, whose boundary lengths sum to twice the number of edges, and whose face boundary lengths are each at least three.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys open scoped BigOperators set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.connected_planar_drawing_face_count_certificate__p2m_230ef2ff_d0ad_4b7a_9595_07bea2af44fc_a1 {V : Type} [Fintype V] [LinearOrder V] (L : SimpleGraph V) (hL : HasPlanarDrawingOn L) (hconn : L.Connected) (hcard : 3 ≤ Fintype.card V) : (letI : DecidableRel L.Adj := Classical.decRel L.Adj; ∃ (F : ℕ) (ell : Fin F → ℕ), Fintype.card V + F = L.edgeFinset.card + 2 ∧ (∑ f : Fin F, ell f) = 2 * L.edgeFinset.card ∧ ∀ f, 3 ≤ ell f) := 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).