Exact edge-count planar completion

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WilliamsonShmoys.exists_planar_triangulation_extension__p2m_5e0b2c41_06cd_4cea_b53d_448657bee65e_a1

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

approximation-algorithmsplanar-graphstreewidth

Every connected finite simple graph with at least three vertices and a drawing satisfying HasPlanarDrawingOn extends, on the same vertex set, to a graph with such a drawing and exactly 3|V|−6 edges. The extension contains every original edge. At |V|=3 the exact count is 3, as witnessed by K₃.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_planar_triangulation_extension__p2m_5e0b2c41_06cd_4cea_b53d_448657bee65e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) [DecidableRel G.Adj]
    (hG : HasPlanarDrawingOn G)
    (hcard : 3 ≤ Fintype.card V)
    (hconn : G.Connected) :
    ∃ (H : SimpleGraph V), ∃ (dH : DecidableRel H.Adj),
      HasPlanarDrawingOn H ∧ G ≤ H ∧
        (letI : DecidableRel H.Adj := dH
         H.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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