Maximal planar extension on the same vertices

Proved

WilliamsonShmoys.planar_triangulation_extension__p2m_82dc859a_8068_4b67_a69e_69714826e35e_a1

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Any finite simple connected graph with a crossing-free drawing and at least three vertices has a planar supergraph on the same vertex type whose edge count is exactly three times the number of vertices minus six.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_triangulation_extension__p2m_82dc859a_8068_4b67_a69e_69714826e35e_a1 {V : Type} [Fintype V] [LinearOrder V] (H : SimpleGraph V) (hH : HasPlanarDrawingOn H) (hconn : H.Connected) (hV : 3 <= Fintype.card V) : ∃ K : SimpleGraph V, H.edgeSet ⊆ K.edgeSet ∧ K.edgeSet.ncard = 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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