Extend a finite planar drawing to a triangulation

Proved

WilliamsonShmoys.planar_triangulation_extension__p2m_e3fcbf10_6f41_4741_b7a4_cf721019c4f7_a1

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

approximation-algorithmsplanar-graphstreewidth

Every graph on at least three vertices with a drawing satisfying HasPlanarDrawingOn can be extended on the same vertex type to a drawable graph with exactly 3|V|−6 edges. This is the triangulation-extension step; the parent uses its edge count to contradict the given strict upper bound if the extension were equal to the original graph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_triangulation_extension__p2m_e3fcbf10_6f41_4741_b7a4_cf721019c4f7_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (T : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hT : HasPlanarDrawingOn T) :
    ∃ U : SimpleGraph V,
      T ≤ U ∧ HasPlanarDrawingOn U ∧
        (letI : DecidableRel U.Adj := Classical.decRel U.Adj
         U.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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