Planar edge bound for connected drawings
ProvedWilliamsonShmoys.planar_edge_count_connected__p2m_bd57f3c8_e371_4c77_9b1f_383221a33d80_a3approximation-algorithmsplanar-graphstreewidth
This is the connected case needed by the parent split. It removes the connected branch’s edge-count obligation. Prove it from the drawing by constructing a plane embedding, applying Euler’s formula and triangular-face counting after maximal completion, and using edge containment. The |V| >= 3 condition handles the small-vertex boundary where the usual triangular-face argument needs its nondegenerate case. The remaining core is formalizing the embedding and face counts from the arc data.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_edge_count_connected__p2m_bd57f3c8_e371_4c77_9b1f_383221a33d80_a3
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hcard : 3 <= Fintype.card V)
(hconn : K.Connected)
(hK : HasPlanarDrawingOn K) :
letI : (L : SimpleGraph V) -> DecidableRel L.Adj :=
fun L => Classical.decRel L.Adj
K.edgeFinset.card <= 3 * Fintype.card V - 6 := 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).