Euler certificate from a finite planar drawing
OpenWilliamsonShmoys.planar_drawing_euler_certificate__p2m_10555cd3_1e32_4780_a0c9_e859a324fd87_a1approximation-algorithmsplanar-graphstreewidth
A graph with the given crossing-free drawing has a planar simple supergraph on the same vertices that is maximal under adding edges. For at least three vertices, such a maximal planar graph is connected and every face is triangular. Its edge and face counts satisfy Euler’s relation and the face-degree inequality, while its edge count bounds the original graph’s edge count. The augmentation and counting argument must also handle an initially disconnected graph and the three-vertex boundary case.
Preamble
import Mathlib import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_drawing_euler_certificate__p2m_10555cd3_1e32_4780_a0c9_e859a324fd87_a1
{U : Type} [Fintype U] [LinearOrder U]
(K : SimpleGraph U)
(hdraw : WilliamsonShmoys.HasPlanarDrawingOn K) :
∃ m' f : Nat,
K.edgeSet.ncard ≤ m' ∧
Fintype.card U + f = m' + 2 ∧
3 * f ≤ 2 * m' := 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).