Edge count is monotone under graph inclusion

Proved

WilliamsonShmoys.subgraph_edge_card_le__p2m_078d72d1_c4b8_4d42_8ead_9d2688891836_a1

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

approximation-algorithmsplanar-graphstreewidth

For finite graphs on the same vertex type, inclusion of graphs implies that the first graph’s edge count is no larger than the second’s.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.subgraph_edge_card_le__p2m_078d72d1_c4b8_4d42_8ead_9d2688891836_a1 {V : Type} [Fintype V] [LinearOrder V] (G F : SimpleGraph V) [DecidableRel G.Adj] [DecidableRel F.Adj] (hGF : G <= F) : G.edgeFinset.card <= F.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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