Connected planar supergraph on the same vertices
OpenWilliamsonShmoys.planar_connected_supergraph__p2m_f0f39874_5ae5_48cf_ae7d_0d6801141c64_a1approximation-algorithmsplanar-graphstreewidth
Every finite simply drawn planar graph can be extended, using only its existing vertices, to a connected graph that still has a crossing-free drawing.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_connected_supergraph__p2m_f0f39874_5ae5_48cf_ae7d_0d6801141c64_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) [DecidableRel H.Adj]
(hH : HasPlanarDrawingOn H) :
∃ K : SimpleGraph V,
(∀ u v, H.Adj u v → K.Adj u v) ∧
K.Connected ∧ HasPlanarDrawingOn K := by sorrySource
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).