Triangulating a drawable graph on the same vertices

Proved

WilliamsonShmoys.planar_supergraph_triangulation__p2m_91b0396d_b61c_4659_90f8_b13543ceeb7a_a1

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

approximation-algorithmsplanar-graphstreewidth

Every graph on a finite linearly ordered vertex type with at least three vertices and a drawing satisfying HasPlanarDrawingOn is contained in a graph on the same vertex type that also satisfies HasPlanarDrawingOn and has exactly 3|V|−6 edges. This is the standard planar triangulation-extension result, specialized to the supplied drawing predicate; its proof must preserve all vertices and the precise crossing-free arc conditions.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_supergraph_triangulation__p2m_91b0396d_b61c_4659_90f8_b13543ceeb7a_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hKDrawing : HasPlanarDrawingOn K) :
    letI : ∀ L : SimpleGraph V, DecidableRel L.Adj := fun L => Classical.decRel L.Adj
    ∃ L : SimpleGraph V, K ≤ L ∧ HasPlanarDrawingOn L ∧
      L.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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