Exact edge-count planar completion

Proved

WilliamsonShmoys.planar_triangulation_supergraph_exact_edge_count__p2m_870fe399_b049_44c0_b9a7_7936a0914e85_a2

by Eyal1990 · Oct 31, 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 3n−6 edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_triangulation_supergraph_exact_edge_count__p2m_870fe399_b049_44c0_b9a7_7936a0914e85_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (J : SimpleGraph V)
    (hJ : WilliamsonShmoys.HasPlanarDrawingOn J)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
      J ≤ K ∧ WilliamsonShmoys.HasPlanarDrawingOn K ∧
        (letI : DecidableRel K.Adj := dK;
          3 * Fintype.card V - 6 = K.edgeFinset.card) := 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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