Component edge bounds

Proved

WilliamsonShmoys.component_edge_bounds__p2m_ddc73965_e52c_4f89_a584_0b5c76884424_a1__p2m_ddc73965_e52c_4f89_a584_0b5c76884424_a3

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For each component support of size at most two, simplicity bounds the number of induced edges by one. For larger supports, the induced graph is connected, so the supplied local hypothesis gives the stated edge bound. The thresholds and additive constant are unchanged.

Preamble

import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.component_edge_bounds__p2m_ddc73965_e52c_4f89_a584_0b5c76884424_a1__p2m_ddc73965_e52c_4f89_a584_0b5c76884424_a3 {V : Type} [Fintype V] [LinearOrder V] (L : SimpleGraph V) (hLocal : ∀ (S : Finset V), 3 ≤ S.card → (L.induce (↑S : Set V)).Connected → (L.induce (↑S : Set V)).edgeSet.ncard + 6 ≤ 3 * S.card) : (∀ c : L.ConnectedComponent, (SimpleGraph.ConnectedComponent.supp c).ncard ≤ 2 → (L.induce (SimpleGraph.ConnectedComponent.supp c)).edgeSet.ncard ≤ 1) ∧ (∀ c : L.ConnectedComponent, 3 ≤ (SimpleGraph.ConnectedComponent.supp c).ncard → (L.induce (SimpleGraph.ConnectedComponent.supp c)).edgeSet.ncard + 6 ≤ 3 * (SimpleGraph.ConnectedComponent.supp c).ncard) := 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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