Complete a planar drawing to a triangulation

Proved

WilliamsonShmoys.planar_triangulation_completion__p2m_b1659f81_1f0c_4922_9013_d0dbf115cc22_a2

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph with at least three vertices is contained in a drawable graph on the same vertex set whose edge count is exactly 3|V|−6. This is the finite planar triangulation-completion theorem, with the edge count supplied by Euler’s formula.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_triangulation_completion__p2m_b1659f81_1f0c_4922_9013_d0dbf115cc22_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hL : WilliamsonShmoys.HasPlanarDrawingOn L) :
    ∃ M : SimpleGraph V,
      L ≤ M ∧ WilliamsonShmoys.HasPlanarDrawingOn M ∧
        3 * Fintype.card V - 6 = M.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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