Every drawable graph extends to a triangulated drawable graph

Proved

WilliamsonShmoys.drawable_graph_has_triangulated_extension__p2m_4b104b32_56a0_4b02_b537_5e9d1339dc2d_a2

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite simple graph on at least three vertices has a drawable supergraph on the same vertices with exactly 3|V|−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.drawable_graph_has_triangulated_extension__p2m_4b104b32_56a0_4b02_b537_5e9d1339dc2d_a2 {V : Type} [Fintype V] [LinearOrder V] (P : SimpleGraph V) [DecidableRel P.Adj] (hcard : 3 ≤ Fintype.card V) (hP : HasPlanarDrawingOn P) : ∃ Q : SimpleGraph V, ∃ hQdec : DecidableRel Q.Adj, (letI : DecidableRel Q.Adj := hQdec; P ≤ Q ∧ HasPlanarDrawingOn Q ∧ Q.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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