A drawable graph extends to a triangulation

Proved

WilliamsonShmoys.exists_triangulated_drawable_supergraph__p2m_afa4574a_1607_44bb_93bd_58940dcb1487_a2

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 extends, on the same vertex set, to a drawable graph whose edge count is three times the number of vertices minus six.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_triangulated_drawable_supergraph__p2m_afa4574a_1607_44bb_93bd_58940dcb1487_a2 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) [DecidableRel M.Adj] (hcard : 3 ≤ Fintype.card V) (hM : HasPlanarDrawingOn M) : ∃ N : SimpleGraph V, ∃ (hNdec : DecidableRel N.Adj), M ≤ N ∧ HasPlanarDrawingOn N ∧ (letI : DecidableRel N.Adj := hNdec; N.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).

View graph