Connect a finite planar drawing with at least three vertices
OpenWilliamsonShmoys_planar_connected_extension__p2m_a74ab9cf_0d81_499a_8717_2a45ea195d9c_a3approximation-algorithmsplanar-graphstreewidth
A finite planar drawing on at least three vertices has a connected planar supergraph on the same vertices.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys_planar_connected_extension__p2m_a74ab9cf_0d81_499a_8717_2a45ea195d9c_a3
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
(hcard : 3 ≤ Fintype.card V) :
∃ G : SimpleGraph V, H ≤ G ∧ WilliamsonShmoys.HasPlanarDrawingOn G ∧ G.Connected := 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).