Subgraph equality from edge counts

Proved

WilliamsonShmoys.edge_card_maximality__p2m_934089cc_df1f_4da4_b973_c115b2e35af8_a5

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

approximation-algorithmsplanar-graphstreewidth

For finite edge sets, subgraph inclusion and a reverse inequality on edge counts imply equality of the simple graphs. A proof follows from inclusion of edge finsets and the fact that a strict inclusion of finite sets increases cardinality.

Preamble

import Mathlib

Formal statement

theorem WilliamsonShmoys.edge_card_maximality__p2m_934089cc_df1f_4da4_b973_c115b2e35af8_a5
    {V : Type} [Fintype V]
    {G L : SimpleGraph V} [Fintype G.edgeSet] [Fintype L.edgeSet]
    (hGL : G ≤ L)
    (hcard : L.edgeFinset.card ≤ G.edgeFinset.card) :
    L = G := 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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