Maximal planar graph edge count lower bound
ProvedWilliamsonShmoys.maximal_planar_edge_count_lower__p2m_a82d4d7c_d6e6_4425_a735_0c3a8fa81594_a1approximation-algorithmsplanar-graphstreewidth
A planar graph on at least three vertices that is maximal among planar supergraphs on the same vertices has at least 3|V| − 6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count_lower__p2m_a82d4d7c_d6e6_4425_a735_0c3a8fa81594_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hcard : 3 ≤ Fintype.card V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hmax : ∀ L : SimpleGraph V,
K ≤ L → WilliamsonShmoys.HasPlanarDrawingOn L → L ≤ K) :
3 * Fintype.card V - 6 ≤ K.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).