Planarity is preserved when bridging separate components
OpenWilliamsonShmoys.planar_drawing_add_edge_between_separate_components__p2m_8e313f52_7cd9_446d_9cbb_05951652d57f_a1approximation-algorithmsplanar-graphstreewidth
Given a planar drawing and vertices in distinct connected components, there is a planar drawing of a graph extending the original graph that includes an edge between those vertices.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_add_edge_between_separate_components__p2m_8e313f52_7cd9_446d_9cbb_05951652d57f_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
{u v : V} (hUnreachable : ¬ G.Reachable u v) :
∃ K : SimpleGraph V,
G ≤ K ∧ K.Adj u v ∧ 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).