Euler equation for a maximal planar drawing
ProvedWilliamsonShmoys.maximal_planar_euler_equation__p2m_addb50df_8cce_4ac7_84ef_2e1f35b931aa_a1approximation-algorithmsplanar-graphstreewidth
A maximal drawable simple graph on at least three vertices has a face count satisfying Euler’s equation, with the unbounded face included.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib
Formal statement
theorem WilliamsonShmoys.maximal_planar_euler_equation__p2m_addb50df_8cce_4ac7_84ef_2e1f35b931aa_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V)
(hcard : 3 ≤ Fintype.card V)
(hdraw : WilliamsonShmoys.HasPlanarDrawingOn H)
(hmax : ∀ L : SimpleGraph V,
WilliamsonShmoys.HasPlanarDrawingOn L → H ≤ L → L = H) :
(letI : DecidableRel H.Adj := Classical.decRel H.Adj
∃ 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).