Drawable triangulating supergraph on the same vertices
ProvedWilliamsonShmoys.planar_supergraph_triangulation__p2m_3a2fa668_af0b_4223_89b0_dcabec19097b_a1approximation-algorithmsplanar-graphstreewidth
Every graph drawable in the given sense on at least three finite linearly ordered vertices has a drawable supergraph on the same vertex type with exactly three times the number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_supergraph_triangulation__p2m_3a2fa668_af0b_4223_89b0_dcabec19097b_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V)
(hG : WilliamsonShmoys.HasPlanarDrawingOn G)
(hCard : 3 ≤ Fintype.card V) :
∃ K : SimpleGraph V, G ≤ K ∧ WilliamsonShmoys.HasPlanarDrawingOn K ∧
K.edgeSet.ncard = 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).