Triangulated completion of a finite planar graph

Proved

WilliamsonShmoys.planar_triangulation_completion__p2m_292cccde_9238_49dd_bff4_dbc6d5d73c71_a1

by Eyal1990 · Oct 29, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on a finite ordered vertex type with at least three vertices has a drawable supergraph on the same vertex type whose edge count is exactly three times the number of vertices minus six.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_triangulation_completion__p2m_292cccde_9238_49dd_bff4_dbc6d5d73c71_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hV : 3 ≤ Fintype.card V) :
    ∃ L : SimpleGraph V,
      Set.Subset K.edgeSet L.edgeSet ∧
      WilliamsonShmoys.HasPlanarDrawingOn L ∧
      L.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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