Every drawable graph extends to a triangulation

Proved

maximal_drawable_graph_has_edge_extension_triangulation__p2m_869ffda5_b864_4783_a885_44fd746ff24a_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite linearly ordered vertex type with at least three vertices, a drawable graph with fewer than 3|V|−6 edges has a drawable strict supergraph on the same vertices whose edge count is exactly 3|V|−6.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

Formal statement

theorem maximal_drawable_graph_has_edge_extension_triangulation__p2m_869ffda5_b864_4783_a885_44fd746ff24a_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (dK : DecidableRel K.Adj)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hn : 3 <= Fintype.card V) :
    letI : DecidableRel K.Adj := dK
    K.edgeFinset.card < 3 * Fintype.card V - 6 →
      ∃ L : SimpleGraph V,
        WilliamsonShmoys.HasPlanarDrawingOn L ∧ K ≤ L ∧ ¬ L ≤ K ∧
          (letI : DecidableRel L.Adj := Classical.decRel L.Adj
           L.edgeFinset.card = 3 * Fintype.card V - 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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