Maximal planar edge lower bound
ProvedWilliamsonShmoys.maximal_planar_edge_count_lower__p2m_32302080_d9f8_461c_9542_c9cd7d932e42_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three vertices that is maximal among drawable supergraphs has at least three times the number of vertices minus six edges. The intended proof shows maximality forces all faces in a plane embedding to be triangular, then applies Euler’s formula.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count_lower__p2m_32302080_d9f8_461c_9542_c9cd7d932e42_a1 {V : Type} [Fintype V] [LinearOrder V] (R : SimpleGraph V) (hR : HasPlanarDrawingOn R) (hcard : 3 ≤ Fintype.card V) (hmax : ∀ S : SimpleGraph V, R ≤ S → HasPlanarDrawingOn S → S = R) : ∀ [hRdec : DecidableRel R.Adj], 3 * Fintype.card V - 6 ≤ R.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).