Triangulate a finite drawn simple graph

Proved

planar_drawing_triangulates_to_maximal_planar__p2m_89c022f5_9981_432e_a40e_914f486e5a7e_a1

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

approximation-algorithmsplanar-graphstreewidth

Every crossing-free drawing of a finite simple graph on at least three vertices can be extended, without adding vertices, to a crossing-free simple graph with exactly 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

Formal statement

theorem planar_drawing_triangulates_to_maximal_planar__p2m_89c022f5_9981_432e_a40e_914f486e5a7e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (dK : DecidableRel K.Adj)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hn : 3 <= Fintype.card V) :
    ∃ (K' : SimpleGraph V) (dK' : DecidableRel K'.Adj),
      K ≤ K' ∧ WilliamsonShmoys.HasPlanarDrawingOn K' ∧
        (letI : DecidableRel K'.Adj := dK';
          K'.edgeFinset.card = 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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