Planar triangulation extension with exact edge count

Proved

WilliamsonShmoys.planar_triangulation_extension_edge_count__p2m_92179202_75fd_4ebd_82a2_e854a66d92bd_a2

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph with at least three vertices has a drawable supergraph on the same vertices with exactly 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_triangulation_extension_edge_count__p2m_92179202_75fd_4ebd_82a2_e854a66d92bd_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V)
    (hL : WilliamsonShmoys.HasPlanarDrawingOn L)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ T : SimpleGraph V,
      L.edgeSet ⊆ T.edgeSet ∧
      WilliamsonShmoys.HasPlanarDrawingOn T ∧
      T.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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