Maximal planarity forces the triangulated lower bound
ProvedWilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_345280c0_3608_47c5_9d5e_5604ac889a3e_a1approximation-algorithmsplanar-graphstreewidth
A drawable simple graph maximal among drawable supergraphs on the same vertex type, with 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.maximal_planar_edge_lower_bound__p2m_345280c0_3608_47c5_9d5e_5604ac889a3e_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
(hH : HasPlanarDrawingOn H)
(hmax : ∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K → K = H) :
letI : DecidableRel H.Adj := Classical.decRel H.Adj
3 * Fintype.card V ≤ H.edgeFinset.card + 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).