Complete a planar graph to a connected planar supergraph

Disproved

WilliamsonShmoys.connected_planar_supergraph_of_planar__p2m_7851b0dc_47ad_49a3_a8e3_cfe708bc6902_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

Every finite planar graph under the stated drawing convention has a connected planar supergraph on the same vertex type, obtained by adding edges between its components.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.connected_planar_supergraph_of_planar__p2m_7851b0dc_47ad_49a3_a8e3_cfe708bc6902_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (dG : DecidableRel G.Adj)
    (hG : HasPlanarDrawingOn G) :
    ∃ (H : SimpleGraph V) (dH : DecidableRel H.Adj),
      G ≤ 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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