Maximal planar edge lower bound
ProvedWilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_8584d232_b8ae_4ac2_a58e_af76bb6d7665_a3approximation-algorithmsplanar-graphstreewidth
A graph maximal among drawable supergraphs on the same vertex set, with at least three vertices, has at least 3|V|−6 edges. This removes the maximality-based lower-bound direction from the parent. A proof route uses maximality to rule out disconnected components and nontriangular faces, then applies Euler’s formula and face-edge incidence. The remaining core obligation is formalizing those topological consequences for the supplied arc-based drawing definition.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_8584d232_b8ae_4ac2_a58e_af76bb6d7665_a3
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) [DecidableRel N.Adj]
(hN : HasPlanarDrawingOn N)
(hcard : 3 ≤ Fintype.card V)
(hmax : ∀ K : SimpleGraph V, N ≤ K → HasPlanarDrawingOn K → K = N) :
3 * Fintype.card V ≤ N.edgeFinset.card + 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).