Triangulated extension of a finite planar drawing

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WilliamsonShmoys.maximal_planar_triangulation_extension__p2m_9f263aab_78fd_4368_8cb7_ace177486954_a2

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph on at least three vertices is contained, on the same vertex type, in a drawable graph with exactly 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_triangulation_extension__p2m_9f263aab_78fd_4368_8cb7_ace177486954_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hV : 3 <= Fintype.card V)
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G) :
    ∃ H : SimpleGraph V,
      G <= H ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
      (letI : DecidableRel H.Adj := Classical.decRel H.Adj
       H.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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