Sum connected-part edge bounds

Proved

WilliamsonShmoys.component_bounds_sum__p2m_f78b2624_2de8_40ad_9230_053c73815f63_a3

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex partition into nonempty connected induced subgraphs whose edge counts sum to the graph's edge count, the local bounds on parts of size at least three imply the global bound when the graph has at least three vertices.

Preamble

import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.component_bounds_sum__p2m_f78b2624_2de8_40ad_9230_053c73815f63_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (dN : DecidableRel N.Adj)
    (C : Finset (Finset V))
    (hcard : 3 <= Fintype.card V)
    (hpartition : ∀ v : V, ∃! s : Finset V, s ∈ C ∧ v ∈ s)
    (hnonempty : ∀ s ∈ C, s.Nonempty)
    (hconnected : ∀ s ∈ C, (N.induce (s : Set V)).Connected)
    (hedges : N.edgeFinset.card = ∑ s ∈ C, (N.induce (s : Set V)).edgeFinset.card)
    (hlocal : ∀ s : Finset V, 3 <= s.card →
      (N.induce (s : Set V)).Connected →
      (N.induce (s : Set V)).edgeFinset.card <= 3 * s.card - 6) :
    N.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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