Maximal planar graphs attain the triangulation edge count
OpenWilliamsonShmoys.maximal_planar_simple_graph_edge_lower_bound__p2m_438c2037_6b64_401b_a3fa_84e1ccf42d6c_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal finite simple graph with a crossing-free drawing and at least three vertices has at least 3|V|−6 edges. A proof route is to show maximality forces a connected embedding with triangular faces, then apply Euler's formula and count face-edge incidences. This child carries the main maximality-to-triangulation topology obligation.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib
Formal statement
theorem WilliamsonShmoys.maximal_planar_simple_graph_edge_lower_bound__p2m_438c2037_6b64_401b_a3fa_84e1ccf42d6c_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hcard : 3 ≤ Fintype.card V)
(hmax : ∀ K' : SimpleGraph V, K ≤ K' →
WilliamsonShmoys.HasPlanarDrawingOn K' → K' = K) :
3 * Fintype.card V ≤ K.edgeSet.ncard + 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).