Face count and boundary lengths from a planar drawing
ProvedWilliamsonShmoys.planar_drawing_face_count__p2m_00ee0ef8_811a_4a39_b8c0_1ed175b9b4ef_a1approximation-algorithmsplanar-graphstreewidth
A finite connected simple graph on at least three vertices with the specified crossing-free drawing has a number of faces satisfying Euler’s formula and the bound that total face-boundary length is at most twice the number of edges. The drawing-to-plane-embedding and face-boundary argument is the remaining topological core.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_face_count__p2m_00ee0ef8_811a_4a39_b8c0_1ed175b9b4ef_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) [DecidableRel K.Adj]
(hV : 3 ≤ Fintype.card V)
(hconn : K.Connected)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
∃ 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).