Planar triangulation extension edge count
ProvedWilliamsonShmoys.planar_triangulation_extension_edge_count__p2m_d4833247_a107_430f_bfde_1b2cd697853a_a1approximation-algorithmsplanar-graphstreewidth
Every drawable finite simple graph on at least three vertices has a drawable supergraph on the same vertex type whose edge count is exactly three times the number of vertices minus six.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_triangulation_extension_edge_count__p2m_d4833247_a107_430f_bfde_1b2cd697853a_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V)
(hV : 3 ≤ Fintype.card V)
(hdraw : WilliamsonShmoys.HasPlanarDrawingOn H) :
∃ T : SimpleGraph V,
Set.Subset H.edgeSet T.edgeSet ∧
WilliamsonShmoys.HasPlanarDrawingOn T ∧
T.edgeSet.ncard + 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).