Edge count is bounded by a triangulated supergraph

Proved

WilliamsonShmoys.edge_count_le_of_triangulated_supergraph__p2m_dc457ca2_cc5d_4b3b_81b3_dc716865aa77_a1

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

approximation-algorithmsplanar-graphstreewidth

If the edges of a finite simple graph are included among those of a graph on the same vertices whose edge count is exactly 3|V|−6, then the first graph has at most 3|V|−6 edges. The exact cardinality hypothesis handles the subtraction boundary directly; the parent supplies at least three vertices.

Preamble

import Mathlib

Formal statement

theorem WilliamsonShmoys.edge_count_le_of_triangulated_supergraph__p2m_dc457ca2_cc5d_4b3b_81b3_dc716865aa77_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G H : SimpleGraph V) (dG : DecidableRel G.Adj) (dH : DecidableRel H.Adj)
    (hsub : (letI : DecidableRel G.Adj := dG; letI : DecidableRel H.Adj := dH; G.edgeFinset ⊆ H.edgeFinset))
    (hcard : (letI : DecidableRel H.Adj := dH; H.edgeFinset.card = 3 * Fintype.card V - 6)) :
    (letI : DecidableRel G.Adj := dG; G.edgeFinset.card ≤ 3 * Fintype.card V - 6) := 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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