Edge count of a maximal drawable graph
ProvedWilliamsonShmoys.maximal_planar_edge_count__p2m_6d98c2a3_7d62_42a8_a794_93d41b1b40ef_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three vertices that has no drawable strict supergraph on the same vertices has exactly 3|V|−6 edges. The proof route is to show maximality forces every face to be triangular, then apply Euler’s formula.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_6d98c2a3_7d62_42a8_a794_93d41b1b40ef_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hK : HasPlanarDrawingOn K)
(hmax : ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L ≤ K)
(hV : 3 ≤ Fintype.card V) :
letI : DecidableRel K.Adj := Classical.decRel K.Adj
letI : Fintype ↥K.edgeSet := Fintype.ofFinite _
K.edgeFinset.card + 6 = 3 * Fintype.card V := 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).