Connect a finite planar drawing by adding edges
ProvedWilliamsonShmoys.planar_drawing_connected_supergraph__p2m_e0c85ed3_29c9_4753_86a9_a6c3e4be15ff_a2approximation-algorithmsplanar-graphstreewidth
A finite nonempty graph with a crossing-free drawing can be extended, on the same vertex set, to a connected graph that contains all its original edges and still has a crossing-free drawing.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_connected_supergraph__p2m_e0c85ed3_29c9_4753_86a9_a6c3e4be15ff_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V)
(hcard : 0 < Fintype.card V)
(hdraw : WilliamsonShmoys.HasPlanarDrawingOn H) :
∃ K : SimpleGraph V, H ≤ K ∧ K.Connected ∧ WilliamsonShmoys.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).