Monotonicity of finite graph edge counts

Proved

WilliamsonShmoys.edgeFinset_card_mono__p2m_ad7657d2_129c_4ff8_bf1d_16d96be983d7_a1

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

approximation-algorithmsplanar-graphstreewidth

If every edge of one finite simple graph is an edge of another graph on the same vertex type, then its edge count is no larger.

Preamble

import Mathlib

Formal statement

theorem WilliamsonShmoys.edgeFinset_card_mono__p2m_ad7657d2_129c_4ff8_bf1d_16d96be983d7_a1
    {V : Type} [Fintype V]
    (H G : SimpleGraph V) [DecidableRel H.Adj] [DecidableRel G.Adj]
    (hHG : H ≤ G) :
    H.edgeFinset.card ≤ G.edgeFinset.card := 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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