Connected planar drawing edge upper bound
ProvedWilliamsonShmoys.planar_drawing_edge_upper_bound_connected_case__p2m_263e0430_0bed_4695_a55f_33a5be744a5b_a1approximation-algorithmsplanar-graphstreewidth
A connected finite simple graph with at least three vertices and a drawing satisfying HasPlanarDrawingOn has at most 3|V| − 6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_drawing_edge_upper_bound_connected_case__p2m_263e0430_0bed_4695_a55f_33a5be744a5b_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) (hV : 3 <= Fintype.card V)
(hconn : N.Connected) (hN : HasPlanarDrawingOn N) :
(letI : DecidableRel N.Adj := Classical.decRel N.Adj;
N.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).