Maximal connected drawable graphs meet the planar edge bound
ProvedWilliamsonShmoys.edgeMaximalPlanar_edgeCount_lower__p2m_4c527512_1622_4d01_a868_d6bc284410d2_a1approximation-algorithmsplanar-graphstreewidth
A connected edge-maximal drawable simple graph on at least three vertices has at least 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.edgeMaximalPlanar_edgeCount_lower__p2m_4c527512_1622_4d01_a868_d6bc284410d2_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) [DecidableRel K.Adj]
(hK : HasPlanarDrawingOn K)
(hmax : ∀ H : SimpleGraph V, K ≤ H → HasPlanarDrawingOn H → H = K)
(hconn : K.Connected)
(hn : 3 ≤ Fintype.card V) :
3 * Fintype.card V - 6 ≤ K.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).