Drawable graphs extend to a triangulation

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WilliamsonShmoys.planar_completion__p2m_31190146_93e5_4043_a404_538cf81f5878_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on a finite linearly ordered vertex type with at least three vertices is contained in a drawable graph on the same vertices whose edge count is exactly three times the vertex count minus six.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_completion__p2m_31190146_93e5_4043_a404_538cf81f5878_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hplanar : HasPlanarDrawingOn L) :
    ∃ K : SimpleGraph V, L.edgeSet ⊆ K.edgeSet ∧
      HasPlanarDrawingOn K ∧
      K.edgeSet.ncard + 6 = 3 * Fintype.card V := 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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