Maximal drawable graph edge-count lower bound
ProvedWilliamsonShmoys.maximal_drawable_graph_edge_count_lower_bound__p2m_267a842e_101e_4cf6_b705_d11d92433d33_a1approximation-algorithmsplanar-graphstreewidth
If a drawable finite simple graph on at least three vertices has no proper drawable supergraph on the same vertices, it has at least 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib
Formal statement
theorem WilliamsonShmoys.maximal_drawable_graph_edge_count_lower_bound__p2m_267a842e_101e_4cf6_b705_d11d92433d33_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (dH : DecidableRel H.Adj)
(hH : WilliamsonShmoys.HasPlanarDrawingOn H)
(hn : 3 <= Fintype.card V)
(hmax : ∀ K : SimpleGraph V,
WilliamsonShmoys.HasPlanarDrawingOn K → H ≤ K → K = H) :
(letI : DecidableRel H.Adj := dH;
3 * Fintype.card V - 6 <= H.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).