Edge bound for maximal drawable graphs
DisprovedWilliamsonShmoys.maximal_planar_drawable_edge_bound__p2m_4f14f4e8_9a22_49bc_9255_581b0400575d_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three vertices that is maximal under drawable edge inclusion has at least 3|V|−6 edges. This separates the Euler counting argument from the completion lemma. A proof route is to show maximality forces a plane embedding whose faces are triangles, then count edge incidences across faces and apply Euler's formula. The hypotheses retain the vertex lower bound and maximality needed for that count.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_drawable_edge_bound__p2m_4f14f4e8_9a22_49bc_9255_581b0400575d_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) [DecidableRel N.Adj]
(hN : HasPlanarDrawingOn N)
(hcard : 3 ≤ Fintype.card V)
(hmax : ∀ K : SimpleGraph V, HasPlanarDrawingOn K → N ≤ K → K ≤ N → N = K) :
3 * Fintype.card V - 6 ≤ N.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).