Equal-or-smaller edge count forces reverse inclusion

Proved

WilliamsonShmoys.edge_cardinality_reflects_inclusion__p2m_04c04e16_b235_4c68_889f_e3b37d35b24b_a1

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

approximation-algorithmsplanar-graphstreewidth

For finite simple graphs, if every edge of K is an edge of L and K has at least as many edges as L, then every edge of L is already in K.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.edge_cardinality_reflects_inclusion__p2m_04c04e16_b235_4c68_889f_e3b37d35b24b_a1
    {V : Type} [Fintype V]
    (K L : SimpleGraph V)
    (hKL : ∀ ⦃u v : V⦄, K.Adj u v → L.Adj u v)
    (hcount : K.edgeSet.ncard ≥ L.edgeSet.ncard) :
    ∀ u v, L.Adj u v → K.Adj u v := 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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