Every finite drawable graph extends to a triangulation

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WilliamsonShmoys.maximal_drawable_graph_has_triangulation__p2m_4eb14725_e37b_413a_b3b4_00357af41eb8_a1

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

approximation-algorithmsplanar-graphstreewidth

Any drawable finite simple graph on 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.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_drawable_graph_has_triangulation__p2m_4eb14725_e37b_413a_b3b4_00357af41eb8_a1 {V : Type} [Fintype V] [LinearOrder V] (N : SimpleGraph V) [DecidableRel N.Adj] (hcard : 3 ≤ Fintype.card V) (hN : HasPlanarDrawingOn N) : ∃ P : SimpleGraph V, N ≤ P ∧ HasPlanarDrawingOn P ∧ ∀ [DecidableRel P.Adj], P.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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