Monotonicity of edge-set cardinality under adjacency inclusion

Proved

WilliamsonShmoys.edgeSet_ncard_mono__p2m_b2bd6a42_fcd9_4968_a08e_3c940e955daf_a1

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

approximation-algorithmsplanar-graphstreewidth

For finite graphs on the same vertex type, adjacency inclusion induces inclusion of edge sets and hence monotonicity of their cardinalities. Prove by showing G.edgeSet ⊆ H.edgeSet and using finite-set cardinality monotonicity.

Preamble

import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.edgeSet_ncard_mono__p2m_b2bd6a42_fcd9_4968_a08e_3c940e955daf_a1
    {V : Type} [Fintype V] [DecidableEq V]
    (G H : SimpleGraph V)
    (hsub : ∀ ⦃u v : V⦄, G.Adj u v → H.Adj u v) :
    G.edgeSet.ncard ≤ H.edgeSet.ncard := 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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