Aggregate edge bounds over connected induced subgraphs

Proved

WilliamsonShmoys.componentwise_edge_bound_aggregate__p2m_c4d63dfe_c88f_43bb_a642_10d75b5a8453_a2

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

approximation-algorithmsplanar-graphstreewidth

If every connected induced subgraph on at least three vertices satisfies the Euler edge bound, then the whole graph does too, provided the whole graph has at least three vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.componentwise_edge_bound_aggregate__p2m_c4d63dfe_c88f_43bb_a642_10d75b5a8453_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hlocal : ∀ S : Finset V, 3 ≤ S.card →
      (K.induce (↑S : Set V)).Connected →
      (K.induce (↑S : Set V)).edgeSet.ncard ≤ 3 * S.card - 6) :
    K.edgeSet.ncard ≤ 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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