Euler face count for a connected planar drawing
ProvedWilliamsonShmoys.planar_drawing_euler_characteristic__p2m_b664cddc_4710_40ab_ab7d_a96b7cb12f1c_a2approximation-algorithmsplanar-graphstreewidth
A finite connected graph with the specified crossing-free drawing has a face count satisfying Euler’s vertex-edge-face equation. Connectedness is needed for this form of the equation; no minimum vertex count is needed.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_drawing_euler_characteristic__p2m_b664cddc_4710_40ab_ab7d_a96b7cb12f1c_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) [DecidableRel K.Adj]
(hconn : K.Connected)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
∃ f : ℕ, Fintype.card V + f = K.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).