Euler formula for a connected planar drawing
ProvedWilliamsonShmoys.connected_planar_euler_formula__p2m_034f5ef2_6768_4eb9_83c3_8c1c8d5ce02b_a2approximation-algorithmsplanar-graphstreewidth
A finite connected graph with a drawing satisfying HasPlanarDrawingOn has a natural face count satisfying Euler’s vertex-edge-face equation.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.connected_planar_euler_formula__p2m_034f5ef2_6768_4eb9_83c3_8c1c8d5ce02b_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) [DecidableRel H.Adj]
(hH : HasPlanarDrawingOn H)
(hconn : H.Connected) :
∃ f : ℕ, Fintype.card V + f = H.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).