Connected planar triangulation completion

Proved

WilliamsonShmoys.connected_planar_triangulation_completion__p2m_571daa92_bad4_4408_bc3e_99c45f7406bd_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every finite simple graph with a crossing-free drawing and at least three vertices has a connected planar supergraph on the same vertices that still has a crossing-free drawing, contains at least the original edge count, and has exactly 3n−6 edges. The hypothesis n ≥ 3 is needed for the triangulation edge formula and includes the boundary case n = 3.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.connected_planar_triangulation_completion__p2m_571daa92_bad4_4408_bc3e_99c45f7406bd_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    (hcard : 3 <= Fintype.card V) :
    letI : DecidableRel G.Adj := Classical.decRel _
    ∃ H : SimpleGraph V,
      (letI : DecidableRel H.Adj := Classical.decRel _;
       G ≤ H ∧ H.Connected ∧
       WilliamsonShmoys.HasPlanarDrawingOn H ∧
       G.edgeFinset.card ≤ H.edgeSet.ncard ∧
       H.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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