Counting blocks in a connected-component partition

Proved

WilliamsonShmoys.componentPartition_count__p2m_954faa17_a8ce_4e0c_b281_2379f22f5ab2_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite graph with at least three vertices, any nonempty disjoint cover by blocks inducing connected subgraphs has at least two blocks if the graph is disconnected, and the sum of the block sizes equals the number of vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open scoped BigOperators

Formal statement

theorem WilliamsonShmoys.componentPartition_count__p2m_954faa17_a8ce_4e0c_b281_2379f22f5ab2_a1
    {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))
    (hnonempty : ∀ s ∈ C, s.Nonempty)
    (hcover : ∀ v : V, ∃ s ∈ C, v ∈ s)
    (hdisjoint : ∀ s ∈ C, ∀ t ∈ C, s ≠ t → Disjoint s t)
    (hconnected : ∀ s ∈ C, (L.induce (↑s : Set V)).Connected) :
    letI : DecidableRel L.Adj := dL
    2 ≤ C.card ∧ (∑ s ∈ C, s.card) = Fintype.card 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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