Extend a planar drawing to a triangulation

Proved

WilliamsonShmoys.planar_drawing_triangulation_extension__p2m_7802999e_536b_4f03_9f51_584d07594d0b_a1

by Eyal1990 · Nov 1, 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 is a subgraph of a drawable graph on the same vertices with exactly 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_triangulation_extension__p2m_7802999e_536b_4f03_9f51_584d07594d0b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hL : HasPlanarDrawingOn L) :
    ∃ M : SimpleGraph V, L ≤ M ∧ HasPlanarDrawingOn M ∧
      (letI : DecidableRel M.Adj := Classical.decRel M.Adj;
       M.edgeFinset.card + 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).

View graph