Finite nonempty graph has a minimal connected extension

Proved

WilliamsonShmoys.exists_minimal_connected_extension__p2m_9960b19b_62d8_4924_9bd4_14f246de67e1_a4

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For every finite nonempty simple graph, there is a connected supergraph in which deleting any added edge disconnects its endpoints. Nonemptiness is required because Mathlib's connectedness predicate entails that the vertex type is nonempty.

Preamble

import Mathlib

set_option autoImplicit false

open SimpleGraph

Formal statement

theorem WilliamsonShmoys.exists_minimal_connected_extension__p2m_9960b19b_62d8_4924_9bd4_14f246de67e1_a4
    {V : Type} [Fintype V] [LinearOrder V] [Nonempty V]
    (M : SimpleGraph V) :
    ∃ H : SimpleGraph V, M ≤ H ∧ H.Connected ∧
      ∀ u v : V, H.Adj u v → ¬ M.Adj u v →
        ¬ (H.deleteEdges {s(u, v)}).Reachable u v := 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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