Face-count certificate for maximal planar drawings
ProvedWilliamsonShmoys.maximal_planar_face_count_certificate__p2m_42928b97_f6a9_47a4_921a_031377856976_a1approximation-algorithmsplanar-graphstreewidth
A maximal drawable simple graph on at least three finite vertices has a face count satisfying Euler’s formula and the triangular face-side count.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_face_count_certificate__p2m_42928b97_f6a9_47a4_921a_031377856976_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hmax : ∀ H : SimpleGraph V, K ≤ H → WilliamsonShmoys.HasPlanarDrawingOn H → H ≤ K) :
∃ f : ℕ,
Fintype.card V + f = K.edgeSet.ncard + 2 ∧
3 * f = 2 * K.edgeSet.ncard := 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).