Edge count of a maximal drawable graph
ProvedWilliamsonShmoys.maximal_planar_edge_count__p2m_55edef6f_d8c2_4514_97bb_024e1dcb6408_a1approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three vertices that has no strictly larger drawable supergraph has exactly 3|V| − 6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_55edef6f_d8c2_4514_97bb_024e1dcb6408_a1
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V) [DecidableRel M.Adj]
(hM : WilliamsonShmoys.HasPlanarDrawingOn M)
(hcard : 3 ≤ Fintype.card V)
(hmax : ∀ N : SimpleGraph V,
WilliamsonShmoys.HasPlanarDrawingOn N → M ≤ N → N ≤ M) :
M.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).