Extension of a drawable graph to a triangulated planar graph

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WilliamsonShmoys.maximal_planar_drawing_has_triangulated_extension__p2m_055aa8e4_db8a_47ed_b733_513d02bcef62_a1

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

approximation-algorithmsplanar-graphstreewidth

Every graph with a crossing-free drawing on a finite ordered vertex type of size at least three is a subgraph of a drawable graph on the same vertices with exactly 3|V|−6 edges. This includes disconnected graphs and the three-vertex boundary case.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_has_triangulated_extension__p2m_055aa8e4_db8a_47ed_b733_513d02bcef62_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hCard : 3 ≤ Fintype.card V) :
    ∃ N : SimpleGraph V,
      H.edgeSet ⊆ N.edgeSet ∧
      WilliamsonShmoys.HasPlanarDrawingOn N ∧
      N.edgeSet.ncard = 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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