Planar drawing extension to a triangulation
OpenWilliamsonShmoys.planar_triangulation_extension_card__p2m_d9ec561c_5413_404b_9cc9_bfd18ae2ee8f_a1approximation-algorithmsplanar-graphstreewidth
Every connected drawable simple graph on at least three vertices has a drawable supergraph on the same vertex set with exactly 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_triangulation_extension_card__p2m_d9ec561c_5413_404b_9cc9_bfd18ae2ee8f_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hN : HasPlanarDrawingOn N) (hconn : N.Connected) :
∃ P : SimpleGraph V, N ≤ P ∧ HasPlanarDrawingOn P ∧
(letI : DecidableRel P.Adj := Classical.decRel P.Adj
3 * Fintype.card V - 6 = P.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).