Extend a planar drawing to a triangulation
ProvedWilliamsonShmoys.planar_triangulation_extension__p2m_447e7603_6167_4b10_a2c2_ce55b7ca23d6_a1approximation-algorithmsplanar-graphstreewidth
Every drawable finite graph on at least three vertices is a subgraph of a drawable graph on the same vertex type whose edge count is exactly 3|V|−6.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_triangulation_extension__p2m_447e7603_6167_4b10_a2c2_ce55b7ca23d6_a1
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V) (hM : HasPlanarDrawingOn M)
(hV : 3 ≤ Fintype.card V) :
∃ K : SimpleGraph V, M ≤ K ∧ HasPlanarDrawingOn K ∧
(letI : DecidableRel K.Adj := Classical.decRel K.Adj;
3 * Fintype.card V = K.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).