Triangulation completion of a finite planar drawing
OpenWilliamsonShmoys.planarDrawing_has_triangulated_supergraph__p2m_9a9e5c69_ecec_470e_a422_cdf4baf4a7b3_a1approximation-algorithmsplanar-graphstreewidth
Every finite drawable simple graph with at least three vertices has a drawable supergraph on the same vertex set whose edge count is exactly three times the number of vertices minus six.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planarDrawing_has_triangulated_supergraph__p2m_9a9e5c69_ecec_470e_a422_cdf4baf4a7b3_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) [DecidableRel H.Adj]
(hH : HasPlanarDrawingOn H)
(hn : 3 ≤ Fintype.card V) :
∃ (J : SimpleGraph V) (_ : DecidableRel J.Adj),
H ≤ J ∧ HasPlanarDrawingOn J ∧
3 * Fintype.card V - 6 = J.edgeFinset.card := 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).