Edge count is monotone under graph inclusion

Proved

WilliamsonShmoys.edgeFinset_card_mono__p2m_d82f089b_ad0e_432a_8486_b2fabea854db_a2

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

approximation-algorithmsplanar-graphstreewidth

For finite simple graphs on the same vertex type, inclusion of adjacency relations implies that the subgraph's edge count is at most the supergraph's edge count. There is no vertex-count or planarity condition.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.edgeFinset_card_mono__p2m_d82f089b_ad0e_432a_8486_b2fabea854db_a2
    {V : Type} [Fintype V]
    (G H : SimpleGraph V) (hGH : G ≤ H) :
    (letI : DecidableRel G.Adj := Classical.decRel G.Adj
     G.edgeFinset.card) ≤
    (letI : DecidableRel H.Adj := Classical.decRel H.Adj
     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).

View graph