Maximal planar edge-count lower bound
ProvedWilliamsonShmoys.maximal_planar_edge_count_lower__p2m_7b71df04_d9fa_431d_9970_a3959e705ba4_a1approximation-algorithmsplanar-graphstreewidth
A graph with at least three vertices that is maximal among supergraphs drawable on the same vertex type has at least 3 |V| - 6 edges. The proof route is to establish connectedness and triangulation of the complementary faces: any face with more than three boundary edges would permit adding a missing diagonal while preserving the drawing, contradicting maximality. This child does not assume the target equality.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count_lower__p2m_7b71df04_d9fa_431d_9970_a3959e705ba4_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hplanar : HasPlanarDrawingOn H)
(hmax : ∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K → K = H) :
3 * Fintype.card V ≤ H.edgeSet.ncard + 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).