Planar triangulation extension on the same vertex set
OpenWilliamsonShmoys.planar_triangulation_extension__p2m_f073101b_bf91_4d7c_9d90_d062f14a8fca_a1approximation-algorithmsplanar-graphstreewidth
Every connected graph with at least three vertices and a drawing satisfying the supplied predicate can be extended, without adding vertices, to a graph with the same kind of drawing and exactly 3|V|−6 edges. The extension contains every original edge.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_triangulation_extension__p2m_f073101b_bf91_4d7c_9d90_d062f14a8fca_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
(hG : WilliamsonShmoys.HasPlanarDrawingOn G) (hconn : G.Connected) :
∃ H : SimpleGraph V, G ≤ H ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
(letI : DecidableRel H.Adj := Classical.decRel H.Adj
H.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).