Edge count of a maximal drawable graph
ProvedWilliamsonShmoys.maximal_planar_edge_count__p2m_6fbbf826_2147_4da9_aa49_9b6333f29881_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three finite vertices that has no strict drawable supergraph on the same vertices has exactly 3|V|−6 edges. The proof must show maximality entails a triangulation and then use Euler’s formula; the at-least-three hypothesis preserves the small-boundary cases.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_6fbbf826_2147_4da9_aa49_9b6333f29881_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) [DecidableRel K.Adj]
(hK : HasPlanarDrawingOn K)
(hmax : ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L = K)
(hcard : 3 ≤ Fintype.card V) :
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).