Count edges covered by disjoint blocks exactly once

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edge_partition_count__p2m_9c944e48_3248_4b41_826f_8af0b6f3bcab_a1

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

approximation-algorithmsplanar-graphstreewidth

If pairwise disjoint finite vertex blocks cover the endpoints of every graph edge, summing, over blocks, the number of graph edges with both endpoints in that block counts every graph edge exactly once.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem edge_partition_count__p2m_9c944e48_3248_4b41_826f_8af0b6f3bcab_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (dL : DecidableRel L.Adj)
    (C : Finset (Finset V))
    (hdisj : ∀ s ∈ C, ∀ t ∈ C, s ≠ t → Disjoint s t)
    (hpair : ∀ u v : V, L.Adj u v → ∃ s ∈ C, u ∈ s ∧ v ∈ s) :
    (letI : DecidableRel L.Adj := dL;
    (∑ s ∈ C, {e ∈ L.edgeFinset | e.toFinset ⊆ s}.card) = L.edgeFinset.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).

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