Lower bound from maximal drawable supergraph
ProvedWilliamsonShmoys.maximal_planar_drawing_edge_count_lower_bound__p2m_e087affc_3721_47e3_9583_19a35624e79d_a1approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three vertices that has no proper drawable supergraph on the same vertex type has at least three times its vertex count minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_edge_count_lower_bound__p2m_e087affc_3721_47e3_9583_19a35624e79d_a1
{V : Type} [Fintype V] [LinearOrder V]
(S : SimpleGraph V) (hS : WilliamsonShmoys.HasPlanarDrawingOn S)
(hcard : 3 ≤ Fintype.card V)
(hmax : ∀ T : SimpleGraph V, S ≤ T → WilliamsonShmoys.HasPlanarDrawingOn T → T = S) :
letI : DecidableRel S.Adj := Classical.decRel S.Adj
3 * Fintype.card V ≤ S.edgeFinset.card + 6 := 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).