Sparse drawable graphs admit a strict planar extension
OpenWilliamsonShmoys.planar_sparse_extension__p2m_5770b3e4_2ec8_40d1_b3e9_07b1ee561228_a1approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three fixed vertices whose edge count is strictly below 3|U|−6 admits a strict supergraph on those vertices that also has a crossing-free drawing.
Preamble
import Mathlib import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_sparse_extension__p2m_5770b3e4_2ec8_40d1_b3e9_07b1ee561228_a1
{U : Type} [Fintype U] [LinearOrder U]
(K : SimpleGraph U) (hcard : 3 ≤ Fintype.card U)
(hdraw : WilliamsonShmoys.HasPlanarDrawingOn K)
(hsparse : K.edgeSet.ncard < 3 * Fintype.card U - 6) :
∃ L : SimpleGraph U, K ≤ L ∧ K ≠ L ∧ WilliamsonShmoys.HasPlanarDrawingOn L := 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).