Completion of a drawable graph to a triangulation

Proved

WilliamsonShmoys.planar_drawing_completion__p2m_dbc103f7_2bde_446c_9b3b_46d38358163c_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on a finite ordered vertex type with at least three vertices is a subgraph of a drawable graph on the same vertices whose edge count is exactly three times the number of vertices minus six. The completion also preserves the corresponding edge-count inequality.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_completion__p2m_dbc103f7_2bde_446c_9b3b_46d38358163c_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hH : HasPlanarDrawingOn H) :
    letI : DecidableRel H.Adj := Classical.decRel H.Adj
    ∃ K : SimpleGraph V,
      letI : DecidableRel K.Adj := Classical.decRel K.Adj
      H ≤ K ∧ HasPlanarDrawingOn K ∧
      H.edgeFinset.card ≤ K.edgeFinset.card ∧
      K.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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