Adding an edge across unreachable vertices increases finite edge rank

Proved

WilliamsonShmoys.finite_graph_edge_rank_progress__p2m_9fc762a8_5f43_4315_941e_21ea4c59e3e4_a2

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, if the resulting graph's edge set is the original edge set plus the pair of unreachable vertices, then its edge-set cardinality strictly increases.

Preamble

import Mathlib

Formal statement

theorem WilliamsonShmoys.finite_graph_edge_rank_progress__p2m_9fc762a8_5f43_4315_941e_21ea4c59e3e4_a2
    {V : Type} [Fintype V]
    (G H : SimpleGraph V) (u v : V)
    (hReach : ¬ G.Reachable u v)
    (hEdges : H.edgeSet = G.edgeSet ∪ {Sym2.mk u v}) :
    G.edgeSet.ncard < H.edgeSet.ncard := 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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