Maximality forces the planar edge-count lower bound
ProvedWilliamsonShmoys.maximal_planar_drawing_edge_count_lower_bound__p2m_2ddf171d_07d8_49d9_9393_f959b6989517_a3approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal drawable simple graph on at least three vertices has at least 3|V| − 6 edges, stated without truncated subtraction.
Preamble
import Mathlib import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_edge_count_lower_bound__p2m_2ddf171d_07d8_49d9_9393_f959b6989517_a3
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V)
(hcard : 3 ≤ Fintype.card V)
(hMdraw : HasPlanarDrawingOn M)
(hMmax : ∀ N : SimpleGraph V, M ≤ N →
HasPlanarDrawingOn N → N ≤ M) :
3 * Fintype.card V ≤
(letI : DecidableRel M.Adj := Classical.decRel M.Adj
M.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).