Triangulation extension of a finite planar drawing

Proved

WilliamsonShmoys.planar_drawing_has_triangulation_extension__p2m_f64d1837_5003_4948_9974_ab50e71d546a_a2

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

approximation-algorithmsplanar-graphstreewidth

Every finite simple graph with a crossing-free drawing on an ordered vertex type of size at least three is a subgraph of a crossing-free triangulation on the same vertex type, with exactly 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib.Tactic

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_has_triangulation_extension__p2m_f64d1837_5003_4948_9974_ab50e71d546a_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : HasPlanarDrawingOn K)
    (hV : LE.le 3 (Fintype.card V)) :
    ∃ H : SimpleGraph V,
      K.edgeSet ⊆ H.edgeSet ∧
      HasPlanarDrawingOn H ∧
      H.edgeSet.ncard = 3 * Fintype.card V - 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).

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