Transfer an edge-set bound to the edge finset

Proved

WilliamsonShmoys.edgeSet_ncard_le_to_edgeFinset_card_le__p2m_90141f6c_e00d_474f_bb06_5b30de1c5624_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite graph with a chosen adjacency decision procedure, a bound on the cardinality of its edge set transfers to the cardinality of its edge finset.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

Formal statement

theorem WilliamsonShmoys.edgeSet_ncard_le_to_edgeFinset_card_le__p2m_90141f6c_e00d_474f_bb06_5b30de1c5624_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (dK : DecidableRel K.Adj)
    (hbound : K.edgeSet.ncard ≤ 3 * Fintype.card V - 6) :
    (letI : DecidableRel K.Adj := dK;
     K.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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