Connect components of a planar drawing

Proved

WilliamsonShmoys.connect_planar_drawing__p2m_52d80640_fd16_42f7_b63d_2010f98eda2f_a1

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

approximation-algorithmsplanar-graphstreewidth

Any drawing on a finite ordered vertex type with at least three vertices can be extended to a connected simple graph on the same vertices while retaining a drawing satisfying HasPlanarDrawingOn. The added arcs join components without crossing existing arcs except at shared endpoints.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.connect_planar_drawing__p2m_52d80640_fd16_42f7_b63d_2010f98eda2f_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    ∃ H : SimpleGraph V, K ≤ H ∧ H.Connected ∧
      WilliamsonShmoys.HasPlanarDrawingOn H := 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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