A planar drawing extends to a triangulation

Proved

WilliamsonShmoys.planar_drawing_triangulating_extension__p2m_3c4a1050_73ac_4fb0_b659_1aa33a199ba4_a1

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

approximation-algorithmsplanar-graphstreewidth

Any graph on at least three vertices with a drawing in the specified sense has a drawable supergraph on the same vertices with exactly 3n−6 edges. This is the augmentation-to-triangulation step; the parent uses maximality to identify that supergraph with K.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_triangulating_extension__p2m_3c4a1050_73ac_4fb0_b659_1aa33a199ba4_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn G) :
    letI : ∀ G : SimpleGraph V, DecidableRel G.Adj := fun G => Classical.decRel G.Adj
    ∃ H : SimpleGraph V, G ≤ H ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
      3 * Fintype.card V - 6 = H.edgeFinset.card := 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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