Inclusion and edge count determine a finite graph

Proved

WilliamsonShmoys.graph_eq_of_le_and_ncard__p2m_97b00d21_85c3_4456_b639_3b0e033871b5_a3

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

approximation-algorithmsplanar-graphstreewidth

If one simple graph on a finite vertex type is a subgraph of another and has at least as many edges, then the graphs are equal.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.graph_eq_of_le_and_ncard__p2m_97b00d21_85c3_4456_b639_3b0e033871b5_a3
    {V : Type} [Fintype V]
    {L M : SimpleGraph V}
    (hLM : L ≤ M) (hcard : M.edgeSet.ncard ≤ L.edgeSet.ncard) : M = L := 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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