Planar edge bound in the connected case

Proved

WilliamsonShmoys.planar_edge_bound_connected_case__p2m_d560826c_2a6b_46ec_826e_191525151789_a1

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

approximation-algorithmsplanar-graphstreewidth

For a connected graph with at least three vertices, the local induced-subgraph bound applied to the full vertex set gives the global edge bound.

Preamble

import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_edge_bound_connected_case__p2m_d560826c_2a6b_46ec_826e_191525151789_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V) (dN : DecidableRel N.Adj)
    (hcard : 3 <= Fintype.card V)
    (hlocal : ∀ s : Finset V, 3 <= s.card →
      (N.induce (s : Set V)).Connected →
      (N.induce (s : Set V)).edgeFinset.card <= 3 * s.card - 6)
    (hconn : N.Connected) :
    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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