Maximal drawable graphs have the planar edge count
ProvedWilliamsonShmoys.maximal_drawable_graph_has_planar_edge_count__p2m_0c1d2833_de32_49c5_aebf_88ae0a9260a2_a1approximation-algorithmsplanar-graphstreewidth
A graph on at least three vertices that has a crossing-free drawing and is maximal among drawable supergraphs on the same vertices has exactly 3|V|−6 edges. The vertex-count condition is retained to cover the range where the planar edge formula applies.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_drawable_graph_has_planar_edge_count__p2m_0c1d2833_de32_49c5_aebf_88ae0a9260a2_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) [DecidableRel H.Adj]
(hcard : 3 ≤ Fintype.card V)
(hH : WilliamsonShmoys.HasPlanarDrawingOn H)
(hmax : ∀ K : SimpleGraph V, H ≤ K →
WilliamsonShmoys.HasPlanarDrawingOn K → K = H) :
H.edgeFinset.card + 6 = 3 * Fintype.card V := 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).