Connect components of a finite planar drawing by adding edges
OpenWilliamsonShmoys.planar_drawing_connected_completion__p2m_0951cfca_b350_4b14_85c9_9d9dc14f4b38_a3approximation-algorithmsplanar-graphstreewidth
Extend any finite planar drawing on the same vertex set to a connected planar drawing, retaining every original edge. The classical adjacency decision procedure is installed in the statement before using edgeFinset. For the proof, connect distinct drawn components by adding an arc in a face incident to them, repeating until connected; finiteness makes the process terminate.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_connected_completion__p2m_0951cfca_b350_4b14_85c9_9d9dc14f4b38_a3
{V : Type} [Fintype V] [LinearOrder V]
(J : SimpleGraph V) (hJ : WilliamsonShmoys.HasPlanarDrawingOn J) :
letI : ∀ K : SimpleGraph V, DecidableRel K.Adj :=
fun K => Classical.decRel K.Adj
∃ H : SimpleGraph V,
WilliamsonShmoys.HasPlanarDrawingOn H ∧
(∀ u v : V, Nonempty (H.Walk u v)) ∧
J.edgeFinset ⊆ H.edgeFinset := 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).