Edge count is monotone under simple-graph containment

Proved

WilliamsonShmoys.edge_count_monotone__p2m_56609264_6a9a_4229_a716_b7a8ddd670b3_a1

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

approximation-algorithmsplanar-graphstreewidth

For finite simple graphs on the same vertex type, containment of adjacency relations implies that the smaller graph has no more edges.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.edge_count_monotone__p2m_56609264_6a9a_4229_a716_b7a8ddd670b3_a1
    {V : Type} [Fintype V]
    (L H : SimpleGraph V) (hLH : L ≤ H) :
    letI : DecidableRel L.Adj := Classical.decRel L.Adj
    letI : DecidableRel H.Adj := Classical.decRel H.Adj
    L.edgeFinset.card ≤ H.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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