Restrict augmented drawing data to original edges

Proved

WilliamsonShmoys.augmented_drawing_restricts_to_original__p2m_b73d0dc0_bd45_4d41_a154_c497fe2a40b3_a2

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

approximation-algorithmsplanar-graphstreewidth

Restrict the augmented drawing and crossing hypotheses to original edges. The graph adjacency inclusions supply the drawing data for each original edge and the crossing condition for each distinct pair of original edges.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.augmented_drawing_restricts_to_original__p2m_b73d0dc0_bd45_4d41_a154_c497fe2a40b3_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (u v : V)
    (p : V → ℝ × ℝ) (arc : V → V → ℝ → ℝ × ℝ)
    (hp : Function.Injective p)
    (hAugEdge : ∀ a b, a < b →
      (H ⊔ SimpleGraph.fromEdgeSet {s(u, v)}).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)
    (hAugCross : ∀ a b x y, a < b → x < y →
      (H ⊔ SimpleGraph.fromEdgeSet {s(u, v)}).Adj a b →
      (H ⊔ SimpleGraph.fromEdgeSet {s(u, v)}).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)) :
    (∀ 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)) := 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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