Edges are conserved across disjoint partition blocks

Proved

WilliamsonShmoys.connectedComponent_edge_conservation_disjoint__p2m_1c5fc6d8_f74d_4a49_af67_c6124a04288f_a2

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

approximation-algorithmsplanar-graphstreewidth

For a finite simple graph, if the vertex blocks are pairwise disjoint and every edge has both endpoints in some block, then the sum of the induced-subgraph edge counts equals the graph’s edge count. Disjointness ensures that no edge is counted in more than one block.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.connectedComponent_edge_conservation_disjoint__p2m_1c5fc6d8_f74d_4a49_af67_c6124a04288f_a2
    {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, (L.induce (↑s : Set V)).edgeFinset.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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