Planar augmentation reaches the triangulation edge threshold

Proved

WilliamsonShmoys.planar_triangulation_augmentation__p2m_d657b208_a585_46f8_ac13_1623c7b09177_a2

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For any drawable simple graph on at least three vertices, one can add edges on the same vertex set while preserving planarity until the edge count is at least 3|V|−6. The local instance is scoped as a term-level let, avoiding the prior failed tactic block.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_triangulation_augmentation__p2m_d657b208_a585_46f8_ac13_1623c7b09177_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : HasPlanarDrawingOn G)
    (hcard : 3 <= Fintype.card V) :
    ∃ H : SimpleGraph V,
      G <= H ∧ HasPlanarDrawingOn H ∧
      (letI : DecidableRel H.Adj := Classical.decRel _;
       3 * Fintype.card V - 6 <= H.edgeFinset.card) := 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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