Drawable supergraph with planar edge lower bound
ProvedWilliamsonShmoys.planar_drawable_supergraph_edge_bound__p2m_57c283a7_68a4_4e3d_813f_6e392f4dfebc_a1approximation-algorithmsplanar-graphstreewidth
Every drawable graph on at least three vertices is contained in a drawable graph on the same vertex set with at least 3|V|−6 edges. Choose a drawable supergraph maximal among the finitely many supergraphs on V; adding edges across non-triangular faces and joining components shows that it is a triangulation. Euler’s formula and counting face-edge incidences give the edge bound. This isolates the topological completion and counting argument from the parent’s use of maximality.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_drawable_supergraph_edge_bound__p2m_57c283a7_68a4_4e3d_813f_6e392f4dfebc_a1
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V) [DecidableRel M.Adj]
(hM : HasPlanarDrawingOn M)
(hcard : 3 ≤ Fintype.card V) :
∃ N : SimpleGraph V, HasPlanarDrawingOn N ∧ M ≤ N ∧
∀ [DecidableRel N.Adj], 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).