Edge count of a maximal planar drawing
ProvedWilliamsonShmoys.maximal_planar_edge_count__p2m_5ca561e0_6863_43fe_aa41_85fb9a3351ed_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three vertices that is inclusion-maximal among drawable graphs has exactly 3|V|−6 edges. The independent proof route is to show maximality forces every face in the planar embedding to be triangular, then combine Euler's formula with the face-edge incidence count. The hypothesis 3 ≤ |V| is required for the maximal planar edge formula.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_5ca561e0_6863_43fe_aa41_85fb9a3351ed_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hmax : forall L : SimpleGraph V,
K <= L -> WilliamsonShmoys.HasPlanarDrawingOn L -> L = K)
(hcard : 3 <= Fintype.card V) :
letI : forall G : SimpleGraph V, DecidableRel G.Adj :=
fun G => Classical.decRel G.Adj
K.edgeFinset.card = 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).