Maximal drawable graphs meet the planar edge threshold
OpenWilliamsonShmoys.maximal_drawable_graph_has_planar_edge_lower_bound__p2m_9940682f_4c7e_48c3_97b4_7cba95f9fa9e_a1approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three vertices that has no strictly larger drawable supergraph has at least edges. The independent proof route is to establish that maximality forces a connected triangulation, and then use Euler's formula and face-edge incidence counting. This is the main remaining topological obligation; the inequality's direction and the three-vertex boundary case agree with the standard planar bound.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_drawable_graph_has_planar_edge_lower_bound__p2m_9940682f_4c7e_48c3_97b4_7cba95f9fa9e_a1
{V : Type} [Fintype V] [LinearOrder V]
(L : SimpleGraph V) [DecidableRel L.Adj]
(hL : HasPlanarDrawingOn L)
(hLmax : ∀ M : SimpleGraph V, L < M → ¬ HasPlanarDrawingOn M)
(hV : 3 ≤ Fintype.card V) :
3 * Fintype.card V - 6 ≤ L.edgeFinset.card := 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).