Maximal drawable graph edge lower bound
ProvedWilliamsonShmoys.maximal_planar_edge_card_lower_bound__p2m_26140456_b7e4_4d83_b9c1_39ddc27ba139_a1approximation-algorithmsplanar-graphstreewidth
A drawable graph maximal under drawable supergraphs, with at least three vertices, has at least 3|V| − 6 edges. This removes the maximality and triangulation argument from the parent; prove it by showing that any non-triangular face permits adding a missing edge while preserving the drawing, then apply Euler’s formula to the resulting triangulation.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_card_lower_bound__p2m_26140456_b7e4_4d83_b9c1_39ddc27ba139_a1
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V) [DecidableRel M.Adj]
(hM : WilliamsonShmoys.HasPlanarDrawingOn M)
(hcard : 3 ≤ Fintype.card V)
(hmax : ∀ N : SimpleGraph V,
WilliamsonShmoys.HasPlanarDrawingOn N → M ≤ N → N ≤ M) :
3 * Fintype.card V - 6 ≤ M.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).