Planar triangulation completion

Proved

WilliamsonShmoys.exists_planar_triangulation_completion__p2m_7eac73df_a948_4c33_96ae_30ffe17dadaf_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite simple graph with at least three vertices and a crossing-free drawing has a planar supergraph on the same vertex set whose edge count is exactly 3|V|−6.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_planar_triangulation_completion__p2m_7eac73df_a948_4c33_96ae_30ffe17dadaf_a1 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) (hplanar : WilliamsonShmoys.HasPlanarDrawingOn M) (hcard : 3 <= Fintype.card V) : ∃ M' : SimpleGraph V, M ≤ M' ∧ WilliamsonShmoys.HasPlanarDrawingOn M' ∧ 3 * Fintype.card V = M'.edgeSet.ncard + 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).

View graph