Edge bound for a finite topological planar drawing
ProvedWilliamsonShmoys.planar_edge_count_upper__p2m_88f6e318_47fb_49e3_83e2_88b806f3a0ca_a1approximation-algorithmsplanar-graphstreewidth
Every simple graph with at least three vertices that has a drawing satisfying HasPlanarDrawingOn has at most 3|V| − 6 edges. This is the Euler upper-bound half of the target.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_edge_count_upper__p2m_88f6e318_47fb_49e3_83e2_88b806f3a0ca_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hV : 3 <= Fintype.card V)
(hH : HasPlanarDrawingOn H) :
(letI : DecidableRel H.Adj := Classical.decRel H.Adj;
H.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).