Planar graph triangulation completion

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WilliamsonShmoys.planar_graph_has_maximal_triangulation__p2m_600971ac_633a_45eb_9e62_2fae1db8e0f7_a1

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

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on a finite ordered vertex set with at least three vertices has a drawable supergraph on the same vertices whose edge count is exactly three times the number of vertices minus six. The child packages the count without subtraction as edge count + 6 = 3 * vertex count.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_graph_has_maximal_triangulation__p2m_600971ac_633a_45eb_9e62_2fae1db8e0f7_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) [DecidableRel M.Adj]
    (hcard : 3 ≤ Fintype.card V)
    (hM : WilliamsonShmoys.HasPlanarDrawingOn M) :
    ∃ (N : SimpleGraph V) (_ : DecidableRel N.Adj),
      (∀ u v, M.Adj u v → N.Adj u v) ∧
      WilliamsonShmoys.HasPlanarDrawingOn N ∧
      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).

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