Connected planar supergraph

Proved

WilliamsonShmoys.planar_connected_supergraph__p2m_81eecf30_e96c_499d_b4eb_13964c396f7c_a2

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

approximation-algorithmsplanar-graphstreewidth

A graph with a planar drawing on a finite ordered vertex type of size at least three has a connected planar supergraph on the same vertices.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_connected_supergraph__p2m_81eecf30_e96c_499d_b4eb_13964c396f7c_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    (hcard : 3 <= Fintype.card V) :
    ∃ C : SimpleGraph V,
      G ≤ C ∧ C.Connected ∧ WilliamsonShmoys.HasPlanarDrawingOn C := 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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