Triangulate a planar graph without adding vertices

Proved

WilliamsonShmoys.disconnected_planar_connected_extension_triangulation__p2m_905a9a19_7f8b_412f_9275_cfdcc0758530_a1

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

approximation-algorithmsplanar-graphstreewidth

Every disconnected graph on at least three vertices with the given crossing-free drawing has a connected planar supergraph on the same vertices that contains all its edges and has the edge count of a triangulation.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.disconnected_planar_connected_extension_triangulation__p2m_905a9a19_7f8b_412f_9275_cfdcc0758530_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hN : WilliamsonShmoys.HasPlanarDrawingOn N)
    (hdisconn : ¬ N.Connected) :
    ∃ H : SimpleGraph V, N ≤ H ∧
      (letI : DecidableRel N.Adj := Classical.decRel N.Adj
       letI : DecidableRel H.Adj := Classical.decRel H.Adj
       N.edgeFinset.card ≤ H.edgeFinset.card) ∧
      H.Connected ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
      (letI : DecidableRel H.Adj := Classical.decRel H.Adj
       H.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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