The one-edge extension strictly increases the edge count

Proved

WilliamsonShmoys.one_edge_extension_graph_facts__p2m_8ebe9775_f6af_4eb7_9344_68ad2c5d7599_a4

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

approximation-algorithmsplanar-graphstreewidth

Adding the unordered edge between distinct unreachable vertices gives a supergraph and strictly increases the finite edge count.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.one_edge_extension_graph_facts__p2m_8ebe9775_f6af_4eb7_9344_68ad2c5d7599_a4
    {V : Type} [Fintype V] [LinearOrder V] (N : SimpleGraph V)
    (u v : V) (huv : u ≠ v) (hUnreachable : ¬ N.Reachable u v) :
    let K : SimpleGraph V :=
      N ⊔ SimpleGraph.fromRel (fun a b => a = u ∧ b = v)
    N ≤ K ∧
      (letI : DecidableRel K.Adj := Classical.decRel K.Adj
       letI : DecidableRel N.Adj := Classical.decRel N.Adj
       N.edgeFinset.card < K.edgeFinset.card) := 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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