Planar simple graph edge upper bound
ProvedWilliamsonShmoys.WilliamsonShmoys_maximal_planar_supergraph_edge_count_upper__p2m_126bf562_f451_4e0b_b87e_ef5c21746e7b_a6approximation-algorithmsplanar-graphstreewidth
A drawable finite simple graph on at least three vertices has at most 3|V|−6 edges, whether or not it is connected.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.WilliamsonShmoys_maximal_planar_supergraph_edge_count_upper__p2m_126bf562_f451_4e0b_b87e_ef5c21746e7b_a6
{V : Type} [Fintype V] [LinearOrder V]
(P : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
(hP : HasPlanarDrawingOn P) :
(letI : DecidableRel P.Adj := Classical.decRel P.Adj;
P.edgeFinset.card + 6 ≤ 3 * Fintype.card V) := 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).