Edge-set maximality implies graph equality

Proved

WilliamsonShmoys.le_ncard_edgeSet_eq__p2m_9ce78458_17c3_4c03_93dc_57f21947e1ab_a2

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

approximation-algorithmsplanar-graphstreewidth

For simple graphs on a finite vertex type, subgraph inclusion together with at least as many edges forces equality.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.le_ncard_edgeSet_eq__p2m_9ce78458_17c3_4c03_93dc_57f21947e1ab_a2
    {V : Type} [Fintype V] [DecidableEq V]
    {H K : SimpleGraph V} (hHK : H ≤ K)
    (hcard : Set.ncard H.edgeSet ≥ Set.ncard K.edgeSet) : K = H := 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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