Connected component partition edge accounting

Proved

WilliamsonShmoys.planar_component_partition_accounting__p2m_e7b88b3e_57d2_4e16_879b_dc797a764fee_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite graph has a partition indexed by its vertices whose nonempty pieces are connected components. The pieces are disjoint and cover all vertices, and every edge is counted exactly once among the induced subgraphs on those pieces.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open Finset

Formal statement

theorem WilliamsonShmoys.planar_component_partition_accounting__p2m_e7b88b3e_57d2_4e16_879b_dc797a764fee_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) :
    ∃ C : V → Finset V,
      (∀ i j, i ≠ j → Disjoint (C i) (C j)) ∧
      (Finset.univ.biUnion C = Finset.univ) ∧
      (L.edgeSet.ncard = ∑ i : V, (L.induce (fun v => v ∈ C i)).edgeSet.ncard) := 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