Triangulating planar supergraph on the same vertices

Proved

WilliamsonShmoys.planar_drawing_has_triangulating_supergraph__p2m_d82f089b_ad0e_432a_8486_b2fabea854db_a1

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

approximation-algorithmsplanar-graphstreewidth

Any graph with the stated crossing-free drawing and at least three vertices is a subgraph of a graph on the same vertices that has such a drawing and exactly $3|V|-6$ edges. The vertex lower bound is needed for the triangulation edge formula; no connectedness assumption is imposed.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_has_triangulating_supergraph__p2m_d82f089b_ad0e_432a_8486_b2fabea854db_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hM : HasPlanarDrawingOn M) :
    ∃ H : SimpleGraph V, M ≤ H ∧ 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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