Drawable planar triangulation extension has the exact edge count

Proved

WilliamsonShmoys.planar_triangulation_extension__p2m_c996decd_a3a2_408a_aa7c_25c4f85a8988_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite linearly ordered vertex type with at least three vertices, any graph with a crossing-free drawing has a drawable supergraph on the same vertices whose edge count is exactly 3|V|−6.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open scoped BigOperators

Formal statement

theorem WilliamsonShmoys.planar_triangulation_extension__p2m_c996decd_a3a2_408a_aa7c_25c4f85a8988_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H) :
    ∃ K : SimpleGraph V, H ≤ K ∧
      WilliamsonShmoys.HasPlanarDrawingOn K ∧
      3 * Fintype.card V - 6 = K.edgeSet.ncard := 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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