Maximal drawable graphs attain the planar edge bound
ProvedWilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_76f2d72e_f140_4cd7_bf10_fd0bb7e9203e_a2approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three vertices with no drawable strict supergraph on the same vertices has at least 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open scoped BigOperators
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_76f2d72e_f140_4cd7_bf10_fd0bb7e9203e_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hH : WilliamsonShmoys.HasPlanarDrawingOn H)
(hmax : ∀ K : SimpleGraph V, H ≤ K →
WilliamsonShmoys.HasPlanarDrawingOn K → K = H) :
3 * Fintype.card V - 6 ≤ H.edgeSet.ncard := 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).