Maximal planar graphs have triangulation edge count
ProvedWilliamsonShmoys.maximal_planar_graph_has_triangulation_edge_count__p2m_584708af_d974_4efc_b22f_05859fd9ba8d_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal simple graph admitting the specified planar drawing on a finite linearly ordered vertex type with at least three vertices has exactly three times the number of vertices minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_graph_has_triangulation_edge_count__p2m_584708af_d974_4efc_b22f_05859fd9ba8d_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V)
(hH : WilliamsonShmoys.HasPlanarDrawingOn H)
(hMaximal : ∀ K : SimpleGraph V,
H ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K → K = H)
(hCard : 3 ≤ Fintype.card V) :
H.edgeSet.ncard = 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).