Triangulated extension of a finite drawable graph

Proved

WilliamsonShmoys.triangulate_planar_graph__p2m_0abcdb56_bfcd_4329_9382_36e3242f584a_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on a finite linearly ordered vertex type with at least three vertices has a drawable supergraph on the same vertex type whose edge count is exactly three times the vertex count minus six. The proof should extend the drawing to a triangulation, including when the original graph is disconnected, and count edges using Euler's formula.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.triangulate_planar_graph__p2m_0abcdb56_bfcd_4329_9382_36e3242f584a_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hL : HasPlanarDrawingOn L) :
    ∃ T : SimpleGraph V, L ≤ T ∧ HasPlanarDrawingOn T ∧
      (letI : DecidableRel T.Adj := Classical.decRel T.Adj;
       3 * Fintype.card V = T.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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