Extend a finite planar drawing to a triangulation

Open

WilliamsonShmoys_planar_triangulation_extension__p2m_2d07592e_de05_4bbe_bc2f_be57428238dd_a3

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

approximation-algorithmsplanar-graphstreewidth

Every graph with a drawing satisfying the supplied crossing-free arc definition, on a finite linearly ordered vertex type with at least three vertices, is a subgraph of a planar graph on the same vertex type with exactly 3|V|−6 edges. The parent applies maximality to identify that extension with the original graph, then obtains the lower bound by arithmetic.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys_planar_triangulation_extension__p2m_2d07592e_de05_4bbe_bc2f_be57428238dd_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ L : SimpleGraph V,
      K ≤ L ∧ WilliamsonShmoys.HasPlanarDrawingOn L ∧
        (letI : DecidableRel L.Adj := Classical.decRel L.Adj
         L.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).

View graph