Planar triangulation extension with exact edge count

Proved

WilliamsonShmoys.planar_triangulation_extension__p2m_d59a8e60_bd9e_4339_9c63_4718b5d88d29_a1

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

approximation-algorithmsplanar-graphstreewidth

Any drawable finite simple graph on at least three vertices extends, on the same vertex type, to a drawable supergraph with exactly 3|V|−6 edges. For the parent, maximality forces that extension to equal K.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_triangulation_extension__p2m_d59a8e60_bd9e_4339_9c63_4718b5d88d29_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hK : HasPlanarDrawingOn K) :
    ∃ L : SimpleGraph V, K ≤ L ∧ HasPlanarDrawingOn L ∧
      (letI : DecidableRel L.Adj := Classical.decRel L.Adj;
        L.edgeFinset.card + 6 = 3 * Fintype.card V) := 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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