Graph inclusion and finite edge-set cardinalities

Proved

WilliamsonShmoys.graph_order_card_facts__p2m_78898e31_8773_419c_99b2_d8837842d221_a2

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

approximation-algorithmsplanar-graphstreewidth

For simple graphs on a finite vertex type, graph inclusion implies monotonicity of the edge-set cardinality; if the cardinalities are equal under inclusion, the graphs are equal. The proof uses edge-set inclusion, finite-set cardinality, and graph extensionality. This step does not depend on planarity.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.graph_order_card_facts__p2m_78898e31_8773_419c_99b2_d8837842d221_a2
    {V : Type} [Fintype V]
    (M N : SimpleGraph V) (hMN : M ≤ N) :
    M.edgeSet.ncard ≤ N.edgeSet.ncard ∧
      (M.edgeSet.ncard = N.edgeSet.ncard → M = N) := 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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