Extend a finite simple graph to an exact feasible edge count

Proved

WilliamsonShmoys.simple_graph_edge_count_completion__p2m_3be8fb19_1696_42e0_add6_8cf83fd59ba5_a2

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

approximation-algorithmsplanar-graphstreewidth

If a finite simple graph has at most m edges and m is no greater than the number of unordered pairs of distinct vertices, extend it to a supergraph with exactly m edges.

Preamble

import Mathlib

Formal statement

theorem WilliamsonShmoys.simple_graph_edge_count_completion__p2m_3be8fb19_1696_42e0_add6_8cf83fd59ba5_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (m : Nat)
    (hG : letI : DecidableRel G.Adj := Classical.decRel G.Adj
      G.edgeFinset.card ≤ m)
    (hcap : m ≤ Nat.choose (Fintype.card V) 2) :
    ∃ H : SimpleGraph V,
      G ≤ H ∧
      (letI : DecidableRel G.Adj := Classical.decRel G.Adj
       letI : DecidableRel H.Adj := Classical.decRel H.Adj
       H.edgeFinset.card = m) := 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).

View graph