Straight segments give valid individual edge arcs

Proved

WilliamsonShmoys.small_card_complete_graph_planar_from_embedding_segment_geometry__p2m_8ad4d3a8_78f8_40f6_aebf_b2ce53dffd6a_a4

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

approximation-algorithmsplanar-graphstreewidth

For an edge with distinct placed endpoints, its affine segment is continuous and injective on the closed unit interval and has the required endpoint values. If the endpoints exhaust the vertices, the segment’s open interior avoids every vertex point: equality with either endpoint would contradict injectivity on the closed interval. Prove continuity coordinatewise, use injectivity of p and u < v for segment injectivity, and use endpoint equations plus injectivity for interior avoidance. This packages the geometric obligation for each edge; the parent handles the crossing condition separately.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.small_card_complete_graph_planar_from_embedding_segment_geometry__p2m_8ad4d3a8_78f8_40f6_aebf_b2ce53dffd6a_a4
    {V : Type} [Fintype V] [LinearOrder V]
    (p : V -> Prod Real Real) (hp : Function.Injective p) (u v : V)
    (huv : u < v) (hvertices : forall w : V, Or (w = u) (w = v)) :
    And (ContinuousOn (fun t : Real => (1 - t) • p u + t • p v) (Set.Icc 0 1))
      (And (Set.InjOn (fun t : Real => (1 - t) • p u + t • p v) (Set.Icc 0 1))
        (And ((fun t : Real => (1 - t) • p u + t • p v) 0 = p u)
          (And ((fun t : Real => (1 - t) • p u + t • p v) 1 = p v)
            (forall t : Real, Set.Ioo (0 : Real) 1 t -> forall w : V,
              Not ((fun t : Real => (1 - t) • p u + t • p v) t = p w))))) := 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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