Edge-count bound for connected planar drawings
ProvedWilliamsonShmoys.connected_planar_edge_count_bound__p2m_2c123b8f_1995_4892_9bf1_663a03a79ac0_a1approximation-algorithmsplanar-graphstreewidth
Every connected simple graph with a crossing-free drawing and at least three vertices has at most three times its vertex count minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.connected_planar_edge_count_bound__p2m_2c123b8f_1995_4892_9bf1_663a03a79ac0_a1
{V : Type} [Fintype V] [LinearOrder V]
(L : SimpleGraph V) (hL : HasPlanarDrawingOn L)
(hconn : L.Connected) (hcard : 3 ≤ Fintype.card V) :
(letI : DecidableRel L.Adj := Classical.decRel L.Adj;
L.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).