Extend a drawable graph to a maximal planar edge count

Proved

WilliamsonShmoys.maximal_planar_triangulation_extension__p2m_b4c9393f_ee8c_44c0_9df7_8513af7ec896_a1

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

approximation-algorithmsplanar-graphstreewidth

Every crossing-free drawing on a finite linearly ordered vertex type with at least three vertices extends, on the same vertex set, to a crossing-free drawing whose graph has exactly 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_triangulation_extension__p2m_b4c9393f_ee8c_44c0_9df7_8513af7ec896_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : HasPlanarDrawingOn H)
    (hV : 3 ≤ Fintype.card V) :
    ∃ (K : SimpleGraph V) (_ : DecidableRel K.Adj),
      H.edgeFinset ⊆ K.edgeFinset ∧
      HasPlanarDrawingOn K ∧
      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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