Aggregate component edge bounds for a disconnected graph

Proved

WilliamsonShmoys.disconnected_component_aggregation__p2m_6875723b_f815_435e_a204_8de0d4b60ca9_a3

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

approximation-algorithmsplanar-graphstreewidth

If every connected component satisfies the bound 3|C|−3 and the finite graph is disconnected, then the whole graph satisfies 3|V|−6. Partition vertices and edges by component, sum the component inequalities, and use that disconnectedness gives at least two components. This removes the finite counting and aggregation work from the parent.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.disconnected_component_aggregation__p2m_6875723b_f815_435e_a204_8de0d4b60ca9_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hdisconn : ¬ K.Connected)
    (hcomponents : ∀ c : K.ConnectedComponent,
      letI : Fintype {v : V // v ∈ c.supp} := Fintype.ofFinite _
      letI : DecidableRel ((K.induce c.supp).Adj) := Classical.decRel _
      (K.induce c.supp).edgeFinset.card ≤
        3 * Fintype.card {v : V // v ∈ c.supp} - 3) :
    letI : (L : SimpleGraph V) → DecidableRel L.Adj :=
      fun L => Classical.decRel L.Adj
    K.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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