A planar graph has a triangulated completion on the same vertices

Proved

WilliamsonShmoys.planar_completion_edge_count__p2m_7126cecb_c28a_4db7_af93_ede0b0cbea72_a5

by Eyal1990 · Oct 29, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every finite drawable simple graph on at least three vertices has a drawable supergraph on the same vertices with exactly 3|V|−6 edges. This supplies a completion whose edge count can be compared with the disconnected graph’s bound.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_completion_edge_count__p2m_7126cecb_c28a_4db7_af93_ede0b0cbea72_a5
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    (hV : 3 ≤ Fintype.card V) :
    ∃ K : SimpleGraph V,
      Set.Subset G.edgeSet K.edgeSet ∧
      WilliamsonShmoys.HasPlanarDrawingOn K ∧
      K.edgeSet.ncard + 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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