Connect a finite planar drawing by adding edges

Disproved

WilliamsonShmoys.planar_drawing_connected_extension__p2m_ad7657d2_129c_4ff8_bf1d_16d96be983d7_a1

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

approximation-algorithmsplanar-graphstreewidth

Any finite graph with a crossing-free drawing on the given ordered vertex type has a connected planar supergraph on the same vertex set.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_connected_extension__p2m_ad7657d2_129c_4ff8_bf1d_16d96be983d7_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : HasPlanarDrawingOn H) :
    ∃ G : SimpleGraph V, H ≤ G ∧ G.Connected ∧ HasPlanarDrawingOn G := 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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