Edge count of an edge-maximal drawable graph
ProvedWilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_6f6a4225_6fc3_4c15_be9a_75786ea7ae4b_a2approximation-algorithmsplanar-graphstreewidth
An edge-maximal graph admitting a crossing-free drawing on at least three vertices has at most three times the number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_6f6a4225_6fc3_4c15_be9a_75786ea7ae4b_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V)
(hcard : 3 <= Fintype.card V)
(hH : HasPlanarDrawingOn H)
(hmax : ∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K → K = H) :
letI : (G : SimpleGraph V) -> DecidableRel G.Adj := fun G => Classical.decRel G.Adj
H.edgeFinset.card <= 3 * Fintype.card V - 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).