Edge count is monotone under adjacency inclusion

Proved

WilliamsonShmoys.edgeFinset_card_mono__p2m_cd005786_48c0_47b3_9c83_f93270ee4ea9_a2

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

approximation-algorithmsplanar-graphstreewidth

For finite simple graphs on the same finite linearly ordered vertex type, if every adjacency of G is an adjacency of H, then G has no more edges than H.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.edgeFinset_card_mono__p2m_cd005786_48c0_47b3_9c83_f93270ee4ea9_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G H : SimpleGraph V) (hsub : forall u v, G.Adj u v -> H.Adj u v) :
    letI : forall K : SimpleGraph V, DecidableRel K.Adj := fun K => Classical.decRel K.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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