Planar graph extension to a triangulation

Proved

WilliamsonShmoys.planar_graph_has_triangulating_extension__p2m_dafc8434_c658_484f_b8e8_b76293ec5b82_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every finite planar simple graph on at least three fixed vertices has a planar supergraph on the same vertex set with exactly three times the number of vertices minus six edges.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_graph_has_triangulating_extension__p2m_dafc8434_c658_484f_b8e8_b76293ec5b82_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hplanar : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hcard : 3 <= Fintype.card V) :
    ∃ K : SimpleGraph V,
      H ≤ K ∧
      WilliamsonShmoys.HasPlanarDrawingOn K ∧
      K.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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