Assemble the planar edge bound from connected induced subgraphs

Proved

WilliamsonShmoys.planar_edge_bound_from_connected_induced__p2m_66ade5d4_3842_4532_bbd5_36e43a15d1f5_a1

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

approximation-algorithmsplanar-graphstreewidth

If each connected induced subgraph of size at least three satisfies the planar edge bound, then the entire finite simple graph satisfies it, including when the graph is disconnected.

Preamble

import Mathlib

Formal statement

theorem WilliamsonShmoys.planar_edge_bound_from_connected_induced__p2m_66ade5d4_3842_4532_bbd5_36e43a15d1f5_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hCard : 3 ≤ Fintype.card V)
    (hBound : ∀ (s : Finset V),
      (H.induce (↑s : Set V)).Connected →
      3 ≤ Fintype.card {v // v ∈ s} →
      (H.induce (↑s : Set V)).edgeSet.ncard + 6 ≤
        3 * Fintype.card {v // v ∈ s}) :
    H.edgeSet.ncard + 6 ≤ 3 * Fintype.card V := 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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