Monotonicity of finite graph edge counts

Proved

WilliamsonShmoys.edgeFinset_mono__p2m_f073101b_bf91_4d7c_9d90_d062f14a8fca_a1

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

approximation-algorithmsplanar-graphstreewidth

If one finite simple graph is a subgraph of another on the same ordered vertex type, its finite edge count is no greater. This supplies the parent’s passage from the original graph to its triangulated extension.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.edgeFinset_mono__p2m_f073101b_bf91_4d7c_9d90_d062f14a8fca_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G H : SimpleGraph V) (hGH : G ≤ H) :
    (letI : DecidableRel G.Adj := Classical.decRel G.Adj
     letI : DecidableRel H.Adj := Classical.decRel H.Adj
     G.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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