Disconnected graph requires at least two connected blocks

Proved

WilliamsonShmoys.componentPartition_card_ge_two__p2m_ad9ec9d5_3963_45af_869f_03d998217399_a2

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

approximation-algorithmsplanar-graphstreewidth

For a graph with at least three vertices that is disconnected, any covering family of nonempty blocks whose induced graphs are connected has at least two members.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.componentPartition_card_ge_two__p2m_ad9ec9d5_3963_45af_869f_03d998217399_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (dL : DecidableRel L.Adj)
    (hconn : ¬ L.Connected)
    (hn : 3 ≤ Fintype.card V)
    (C : Finset (Finset V))
    (hne : ∀ s ∈ C, s.Nonempty)
    (hcover : ∀ v : V, ∃ s ∈ C, v ∈ s)
    (hCconn : ∀ s ∈ C, (L.induce (↑s : Set V)).Connected) :
    2 ≤ C.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).

View graph