Planar edge bound from connected induced subgraphs

Proved

WilliamsonShmoys.planar_edge_bound_from_connected_induced_bounds__p2m_ac18a95b_13f8_4a76_980e_021bef74dbe9_a1

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

approximation-algorithmsplanar-graphstreewidth

If every connected induced subgraph on at least three vertices satisfies the planar edge bound, then the whole finite simple graph satisfies it, provided the whole graph has at least three vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_edge_bound_from_connected_induced_bounds__p2m_ac18a95b_13f8_4a76_980e_021bef74dbe9_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) :
    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).

View graph