Planar drawing extends to a connected supergraph

Proved

WilliamsonShmoys.drawable_connected_supergraph__p2m_270023aa_f627_4b53_97e3_8e37f4bb2eb9_a1

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A finite graph with the supplied planar drawing has a connected planar supergraph on the same vertex set. The parent uses the supergraph to apply the connected edge bound, then transfers the result using monotonicity of edgeFinset.card. The vertex-count hypothesis is retained for the connected bound and covers isolated vertices in the construction.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.drawable_connected_supergraph__p2m_270023aa_f627_4b53_97e3_8e37f4bb2eb9_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) [DecidableRel M.Adj]
    (hM : WilliamsonShmoys.HasPlanarDrawingOn M)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ G : SimpleGraph V,
      M ≤ G ∧ G.Connected ∧ WilliamsonShmoys.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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