Planar triangulation extension
ProvedWilliamsonShmoys.planar_triangulation_extension__p2m_b9887391_36ac_4b13_9489_3daf1e406adb_a1approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on a finite linearly ordered vertex type with at least three vertices has a drawable supergraph on the same vertices whose edge count is exactly 3|V|−6.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_triangulation_extension__p2m_b9887391_36ac_4b13_9489_3daf1e406adb_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) (hN : HasPlanarDrawingOn N)
(hV : 3 ≤ Fintype.card V) :
∃ K : SimpleGraph V, N ≤ K ∧ HasPlanarDrawingOn K ∧
(letI : DecidableRel K.Adj := Classical.decRel K.Adj
letI : Fintype ↥K.edgeSet := Fintype.ofFinite _
K.edgeFinset.card + 6 = 3 * Fintype.card V) := 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).