Connect a finite planar drawing without adding vertices

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WilliamsonShmoys.planar_drawing_connected_augmentation__p2m_2cf5c394_8fe3_4b96_8033_bf5d04367b46_a2

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

approximation-algorithmsplanar-graphstreewidth

A finite graph with at least three vertices and a crossing-free drawing has a connected planar supergraph on the same vertex set.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_connected_augmentation__p2m_2cf5c394_8fe3_4b96_8033_bf5d04367b46_a2 {V : Type} [Fintype V] [LinearOrder V] (L : SimpleGraph V) (hL : HasPlanarDrawingOn L) (hcard : 3 ≤ Fintype.card V) : ∃ H : SimpleGraph V, L ≤ H ∧ H.Connected ∧ 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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