Planar triangulation completion on the same vertex set
OpenWilliamsonShmoys.planar_triangulation_completion__p2m_75881c5e_3a5f_4c38_b417_ac584a1894e5_a1approximation-algorithmsplanar-graphstreewidth
Every drawable finite simple graph with at least three vertices is a subgraph of a drawable graph on the same vertex set having exactly 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_triangulation_completion__p2m_75881c5e_3a5f_4c38_b417_ac584a1894e5_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) [DecidableRel N.Adj]
(hN : HasPlanarDrawingOn N)
(hcard : 3 ≤ Fintype.card V) :
∃ K : SimpleGraph V, N ≤ K ∧ HasPlanarDrawingOn K ∧
∃ (d : DecidableRel K.Adj),
(letI : DecidableRel K.Adj := d;
K.edgeFinset.card = 3 * Fintype.card V - 6) := 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).