Edge count of a triangulated planar extension

Open

WilliamsonShmoys.maximal_planar_triangulation_extension_edge_count__p2m_05082c6c_4384_41d5_8a7a_caf40d114b6e_a1

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

approximation-algorithmsplanar-graphstreewidth

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

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_triangulation_extension_edge_count__p2m_05082c6c_4384_41d5_8a7a_caf40d114b6e_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hn : 3 ≤ Fintype.card V) :
    ∃ (K : SimpleGraph V), ∃ (_ : DecidableRel K.Adj),
      WilliamsonShmoys.HasPlanarDrawingOn K ∧
      (∀ u v, H.Adj u v → K.Adj u v) ∧
      K.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).

View graph