Complete a drawable connected graph to a triangulated planar supergraph

Proved

WilliamsonShmoys.planar_completion_edge_bound__p2m_c26a86c1_c11b_4f85_a158_b9e12bec4015_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite connected graph on at least three linearly ordered vertices with a crossing-free drawing has a drawable supergraph on the same vertices whose edge count is exactly three times the number of vertices minus six; the original graph’s edge count is no larger.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_completion_edge_bound__p2m_c26a86c1_c11b_4f85_a158_b9e12bec4015_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) [DecidableRel G.Adj]
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    (hconn : G.Connected) (hcard : 3 <= Fintype.card V) :
    ∃ (H : SimpleGraph V) (hdec : DecidableRel H.Adj),
      G ≤ H ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
      G.edgeFinset.card ≤ H.edgeFinset.card ∧
      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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