Drawable edge augmentation below the planar bound
ProvedWilliamsonShmoys.drawable_edge_augmentation_below_planar_bound__p2m_54520c23_fb0c_40a7_a4c0_ad4b1f033824_a1__p2m_54520c23_fb0c_40a7_a4c0_ad4b1f033824_a2approximation-algorithmsplanar-graphstreewidth
Any drawable simple graph on at least three finite ordered vertices with fewer than 3|V|−6 edges has a drawable strict supergraph on the same vertices. This isolates the geometric augmentation and planar edge-count argument from the target’s short maximality contradiction.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.drawable_edge_augmentation_below_planar_bound__p2m_54520c23_fb0c_40a7_a4c0_ad4b1f033824_a1__p2m_54520c23_fb0c_40a7_a4c0_ad4b1f033824_a2
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V)
(hcard : 3 ≤ Fintype.card V)
(hdraw : WilliamsonShmoys.HasPlanarDrawingOn G)
(hsmall : (letI : DecidableRel G.Adj := Classical.decRel G.Adj; G.edgeFinset.card) < 3 * Fintype.card V - 6) :
∃ K : SimpleGraph V, G ≤ K ∧ K ≠ G ∧ WilliamsonShmoys.HasPlanarDrawingOn K := 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).