Aggregate connected induced-subgraph edge bounds

Proved

WilliamsonShmoys.component_aggregation_edge_bound__p2m_004e5ed5_cd67_44f8_b54c_b7b595ba6dfb_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For a finite simple graph with at least three vertices, bounds of 3|S|−6 for every connected induced subgraph on at least three vertices imply the global 3|V|−6 edge bound. This removes the disconnected-graph bookkeeping from the topological lemma. Prove it by partitioning vertices into connected components, applying the premise to components of size at least three, bounding components of size one or two directly, and summing the component edge counts. The total-size hypothesis is needed for the final arithmetic.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.component_aggregation_edge_bound__p2m_004e5ed5_cd67_44f8_b54c_b7b595ba6dfb_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (U : SimpleGraph V) (dU : DecidableRel U.Adj)
    (hcard : 3 ≤ Fintype.card V)
    (hcomp : ∀ S : Finset V, 3 ≤ S.card →
      (U.induce (↑S : Set V)).Connected →
      (letI : DecidableRel (U.induce (↑S : Set V)).Adj := Classical.decRel _
       (U.induce (↑S : Set V)).edgeFinset.card ≤ 3 * S.card - 6)) :
    (letI : DecidableRel U.Adj := dU
     U.edgeFinset.card ≤ 3 * Fintype.card V - 6) := 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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