Finite strict extensions yield a connected supergraph

Proved

WilliamsonShmoys.finite_graph_extension_connected__p2m_a6ee2644_08f0_4c4d_a139_48d0925fab45_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, if every disconnected Good graph has a strictly larger Good supergraph, then every Good starting graph has a connected Good supergraph containing it. Prove this by choosing a maximal Good supergraph from the finite set of simple graphs.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.finite_graph_extension_connected__p2m_a6ee2644_08f0_4c4d_a139_48d0925fab45_a1
    {V : Type} [Fintype V]
    (Good : SimpleGraph V → Prop)
    (N : SimpleGraph V)
    (hN : Good N)
    (extend : ∀ (G : SimpleGraph V), Good G → ¬ G.Connected →
      ∃ P : SimpleGraph V, G ≤ P ∧ G ≠ P ∧ Good P) :
    ∃ P : SimpleGraph V, N ≤ P ∧ P.Connected ∧ Good P := 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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