Triangulation completion for a planar drawing

Proved

WilliamsonShmoys.planar_triangulation_completion__p2m_689940da_6233_467d_b487_080b0de38a3b_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite linearly ordered vertex type with at least three vertices is a subgraph of a drawable graph on the same vertices whose edge count is 3|V|−6.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_triangulation_completion__p2m_689940da_6233_467d_b487_080b0de38a3b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (P : SimpleGraph V)
    (hP : WilliamsonShmoys.HasPlanarDrawingOn P)
    (hcard : 3 <= Fintype.card V) :
    ∃ Q : SimpleGraph V,
      (∀ {u v : V}, P.Adj u v → Q.Adj u v) ∧
      WilliamsonShmoys.HasPlanarDrawingOn Q ∧
      (letI : DecidableRel Q.Adj := Classical.decRel Q.Adj
       3 * Fintype.card V = Q.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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