Triangulation extension of a finite planar drawing
ProvedWilliamsonShmoys.planar_drawing_triangulation_extension__p2m_71bf5396_ae49_4e3b_9478_7acac339f298_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 vertex set with exactly three times the number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_drawing_triangulation_extension__p2m_71bf5396_ae49_4e3b_9478_7acac339f298_a1
{V : Type} [Fintype V] [LinearOrder V]
(T : SimpleGraph V) (hT : WilliamsonShmoys.HasPlanarDrawingOn T)
(hcard : 3 <= Fintype.card V) :
∃ U : SimpleGraph V, T ≤ U ∧ WilliamsonShmoys.HasPlanarDrawingOn U ∧
(letI : DecidableRel U.Adj := Classical.decRel U.Adj
3 * Fintype.card V = U.edgeFinset.card + 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).