Edge bound inherited by a subgraph

Proved

WilliamsonShmoys.edge_bound_of_subgraph_of_extremal_planar_graph__p2m_6dc85c3d_85f5_4b78_a1c3_d13b079de4de_a1

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

approximation-algorithmsplanar-graphstreewidth

If one finite simple graph is a subgraph of another graph on the same vertices, then an exact edge-count formula for the larger graph implies the corresponding upper bound for the smaller graph.

Preamble

import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.edge_bound_of_subgraph_of_extremal_planar_graph__p2m_6dc85c3d_85f5_4b78_a1c3_d13b079de4de_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M N : SimpleGraph V) (hMN : M ≤ N)
    (hN : (letI : DecidableRel N.Adj := Classical.decRel N.Adj
           N.edgeFinset.card + 6 = 3 * Fintype.card V)) :
    (letI : DecidableRel M.Adj := Classical.decRel M.Adj
     letI : DecidableRel N.Adj := Classical.decRel N.Adj
     M.edgeFinset.card + 6 <= 3 * Fintype.card 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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