Crossing condition for the augmented bridge arc

Proved

WilliamsonShmoys.planar_bridge_augmented_arc_crossing__p2m_3ac38dfe_f4e4_4970_8b72_a722a3a2a331_a2

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For two distinct ordered edges of the union graph, any equality between their augmented arcs at parameters in the unit interval occurs only at endpoints of both arcs.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_bridge_augmented_arc_crossing__p2m_3ac38dfe_f4e4_4970_8b72_a722a3a2a331_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (u v : V)
    (p : V → ℝ × ℝ) (arc : V → V → ℝ → ℝ × ℝ)
    (bridge : ℝ → ℝ × ℝ)
    (hne : u ≠ v) (hnew : ¬ H.Adj u v)
    (hdraw : Function.Injective p ∧
      (∀ a b, a < b → H.Adj a b →
        ContinuousOn (arc a b) (Set.Icc 0 1) ∧
        Set.InjOn (arc a b) (Set.Icc 0 1) ∧
        arc a b 0 = p a ∧ arc a b 1 = p b ∧
        ∀ t ∈ Set.Ioo (0 : ℝ) 1, ∀ w, arc a b t ≠ p w) ∧
      (∀ a b x y, a < b → x < y → H.Adj a b → H.Adj x y →
        (a, b) ≠ (x, y) →
        ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ s ∈ Set.Icc (0 : ℝ) 1,
          arc a b t = arc x y s →
            (t = 0 ∨ t = 1) ∧ (s = 0 ∨ s = 1)))
    (hbridge : ContinuousOn bridge (Set.Icc (0 : ℝ) 1) ∧
      Set.InjOn bridge (Set.Icc 0 1) ∧ bridge 0 = p u ∧ bridge 1 = p v ∧
      (∀ t ∈ Set.Ioo (0 : ℝ) 1, ∀ w, bridge t ≠ p w) ∧
      (∀ a b, a < b → H.Adj a b →
        ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ s ∈ Set.Icc (0 : ℝ) 1,
          bridge t = arc a b s →
            (t = 0 ∨ t = 1) ∧ (s = 0 ∨ s = 1))) :
    ∀ a b x y, a < b → x < y →
      (H ⊔ SimpleGraph.fromEdgeSet ({s(u, v)} : Set (Sym2 V))).Adj a b →
      (H ⊔ SimpleGraph.fromEdgeSet ({s(u, v)} : Set (Sym2 V))).Adj x y →
      (a, b) ≠ (x, y) →
      ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ s ∈ Set.Icc (0 : ℝ) 1,
        (fun a b t => if u < v then
          if a = u ∧ b = v then bridge t else arc a b t
        else if a = v ∧ b = u then bridge (1 - t) else arc a b t) a b t =
        (fun a b t => if u < v then
          if a = u ∧ b = v then bridge t else arc a b t
        else if a = v ∧ b = u then bridge (1 - t) else arc a b t) x y s →
        (t = 0 ∨ t = 1) ∧ (s = 0 ∨ s = 1) := by sorry

Source

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).

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