A drawable graph below the planar edge bound has a drawable strict extension
Provedmaximal_drawable_graph_has_edge_extension__p2m_70fb6007_5dc1_4484_a53d_f6e6d3401839_a1approximation-algorithmsplanar-graphstreewidth
On a finite linearly ordered vertex type with at least three vertices, any drawable graph with fewer than 3|V|−6 edges has a drawable strict supergraph on the same vertices.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib
Formal statement
theorem maximal_drawable_graph_has_edge_extension__p2m_70fb6007_5dc1_4484_a53d_f6e6d3401839_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (dK : DecidableRel K.Adj)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hn : 3 <= Fintype.card V) :
letI : DecidableRel K.Adj := dK
K.edgeFinset.card < 3 * Fintype.card V - 6 →
∃ L : SimpleGraph V,
WilliamsonShmoys.HasPlanarDrawingOn L ∧ K ≤ L ∧ ¬ L ≤ 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).