Triangulate a connected planar drawing without adding vertices

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WilliamsonShmoys_triangulation_supergraph_of_planar_drawing__p2m_02efd696_96fe_4670_9021_ba912a848fde_a2

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

approximation-algorithmsplanar-graphstreewidth

A connected finite simple graph with at least three vertices and a drawing satisfying HasPlanarDrawingOn is contained in a graph on the same vertices with such a drawing and exactly 3 * |V| - 6 edges. The statement provides the classical decidable adjacency instance required for the edge-finset expression.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys_triangulation_supergraph_of_planar_drawing__p2m_02efd696_96fe_4670_9021_ba912a848fde_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hconn : G.Connected)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn G) :
    ∃ H : SimpleGraph V,
      G ≤ H ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
        (letI : DecidableRel H.Adj := Classical.decRel H.Adj
         H.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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