Maximal planar drawing edge lower bound
ProvedWilliamsonShmoys.maximal_planar_drawing_edge_lower_bound__p2m_b022b4b8_e7d5_474e_bb13_2a0e5909c103_a1approximation-algorithmsplanar-graphstreewidth
If a graph on at least three vertices has the specified planar drawing and no drawable supergraph on those vertices properly extends it, then it has at least 3|V|−6 edges. The standard route is to show maximality forces a triangulated embedding and count faces and edge incidences using Euler's formula.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawing_edge_lower_bound__p2m_b022b4b8_e7d5_474e_bb13_2a0e5909c103_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hV : 3 <= Fintype.card V)
(hK : HasPlanarDrawingOn K)
(hmax : ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L ≤ K) :
(letI : DecidableRel K.Adj := Classical.decRel K.Adj;
3 * Fintype.card V - 6 <= K.edgeFinset.card) := 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).