Edge count additivity over a disjoint cover

Disproved

WilliamsonShmoys.component_edge_sum__p2m_43810500_1f43_4f26_9fb2_357a895d104d_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite simple graph, if a finite family of vertex blocks is pairwise disjoint and covers every vertex, then the original edge count equals the sum of the edge counts in the induced subgraphs on those blocks. The cover and disjointness conditions also make the claim valid for an empty vertex type and its empty block family.

Preamble

import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.component_edge_sum__p2m_43810500_1f43_4f26_9fb2_357a895d104d_a1
    {V : Type} [Fintype V] [LinearOrder V] (N : SimpleGraph V)
    (C : Finset (Finset V)) :
    (letI : DecidableRel N.Adj := Classical.decRel N.Adj;
     (∀ B₁ ∈ C, ∀ B₂ ∈ C, B₁ ≠ B₂ →
        Disjoint (↑B₁ : Set V) (↑B₂ : Set V)) →
     (∀ v : V, ∃ B ∈ C, v ∈ B) →
     N.edgeFinset.card =
       ∑ B ∈ C,
         (letI : Fintype {v // v ∈ (↑B : Set V)} := Fintype.ofFinite _;
          (N.induce (↑B : Set V)).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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