Exact edge-count drawable completion

Proved

WilliamsonShmoys.planar_triangulation_supergraph_exact_edge_count__p2m_9fa2b5a7_e492_409e_9c7c_44f6f21de4a2_a3

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

approximation-algorithmsplanar-graphstreewidth

Every drawable simple graph on a finite linearly ordered vertex type with at least three vertices is contained, on that same vertex type, in a drawable graph with exactly 3n − 6 edges. The condition n ≥ 3 covers the triangle boundary case and is needed for the stated exact count.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_triangulation_supergraph_exact_edge_count__p2m_9fa2b5a7_e492_409e_9c7c_44f6f21de4a2_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hG : HasPlanarDrawingOn G) :
    letI : ∀ K : SimpleGraph V, DecidableRel K.Adj :=
      fun K => Classical.decRel K.Adj
    ∃ K : SimpleGraph V,
      G.edgeFinset ⊆ K.edgeFinset ∧
      HasPlanarDrawingOn K ∧
      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).

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