Completion of a drawable graph to a triangulation edge count

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WilliamsonShmoys.planar_completion_edge_count__p2m_b83083f9_5b98_4643_9156_0c35319b256b_a2

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

approximation-algorithmsplanar-graphstreewidth

Any drawable finite simple graph on at least three vertices can be extended on the same vertex set to a drawable graph with exactly three times the number of vertices minus six edges. The child also returns the edge-set finiteness instance required by Mathlib’s edgeFinset definition.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_completion_edge_count__p2m_b83083f9_5b98_4643_9156_0c35319b256b_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hcard : 3 ≤ Fintype.card V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H) :
    ∃ K : SimpleGraph V,
      H ≤ K ∧ WilliamsonShmoys.HasPlanarDrawingOn K ∧
      ∃ (hKfin : Fintype K.edgeSet),
        (letI : Fintype K.edgeSet := hKfin
         3 * Fintype.card V = K.edgeFinset.card + 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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