Finite partition into connected induced blocks

Proved

WilliamsonShmoys.componentPartition_partition__p2m_ad9ec9d5_3963_45af_869f_03d998217399_a2

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

approximation-algorithmsplanar-graphstreewidth

Every finite graph has a partition of its vertices into nonempty blocks whose induced graphs are connected. The blocks cover every vertex, are pairwise disjoint, and their cardinalities sum to the number of vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open scoped BigOperators

Formal statement

theorem WilliamsonShmoys.componentPartition_partition__p2m_ad9ec9d5_3963_45af_869f_03d998217399_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (dL : DecidableRel L.Adj) :
    letI : DecidableRel L.Adj := dL
    ∃ C : Finset (Finset V),
      (∀ s ∈ C, s.Nonempty) ∧
      (∀ v : V, ∃ s ∈ C, v ∈ s) ∧
      (∀ s ∈ C, ∀ t ∈ C, s ≠ t → Disjoint s t) ∧
      (∑ s ∈ C, s.card) = Fintype.card V ∧
      (∀ s ∈ C, (L.induce (↑s : Set V)).Connected) := 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