Extremal planar completion on the same vertex type

Proved

WilliamsonShmoys.planar_drawing_triangulation_extension__p2m_48fd94ac_39d1_4989_ad8e_5c8896a37126_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite drawable simple graph with at least three vertices is contained, on the same vertex type, in a drawable graph with exactly 3|V| − 6 edges. The child isolates the topological completion and triangulation step; the parent uses only its edge-count conclusion.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_triangulation_extension__p2m_48fd94ac_39d1_4989_ad8e_5c8896a37126_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hcard : 3 <= Fintype.card V) :
    letI : forall (K : SimpleGraph V), DecidableRel K.Adj :=
      fun K => Classical.decRel K.Adj
    Exists fun K : SimpleGraph V =>
      H.edgeFinset.card <= K.edgeFinset.card /\
      K.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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