Triangulating a finite planar graph without adding vertices

Proved

WilliamsonShmoys.planarAugmentation_edgeCount__p2m_6ad3e488_df98_4029_b402_def6549b4337_a1

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

approximation-algorithmsplanar-graphstreewidth

Every finite simply drawn planar graph on at least three vertices has a planar supergraph on the same vertices whose faces are triangular and whose edge count is 3|V|−6.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planarAugmentation_edgeCount__p2m_6ad3e488_df98_4029_b402_def6549b4337_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (dK : DecidableRel K.Adj)
    (hK : HasPlanarDrawingOn K)
    (hn : 3 ≤ Fintype.card V) :
    ∃ (L : SimpleGraph V) (dL : DecidableRel L.Adj),
      HasPlanarDrawingOn L ∧
      (letI : DecidableRel K.Adj := dK
       letI : DecidableRel L.Adj := dL
       K.edgeFinset ⊆ L.edgeFinset) ∧
      (letI : DecidableRel L.Adj := dL
       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).

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