Add an edge between two components of a drawable graph
OpenWilliamsonShmoys.planarDrawing_add_edge_of_not_reachable__p2m_889652c1_a82f_462e_b8d8_8391a5f102fc_a2approximation-algorithmsplanar-graphstreewidth
Given a planar drawing and two vertices in different components, add exactly their edge while retaining a planar drawing. The lemma makes the component-joining geometry reusable for any such pair.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib
Formal statement
theorem WilliamsonShmoys.planarDrawing_add_edge_of_not_reachable__p2m_889652c1_a82f_462e_b8d8_8391a5f102fc_a2
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V)
(hG : WilliamsonShmoys.HasPlanarDrawingOn G)
(u v : V)
(huv : ¬ G.Reachable u v) :
∃ H : SimpleGraph V,
G ≤ H ∧
H.edgeSet = G.edgeSet ∪ {Sym2.mk u v} ∧
WilliamsonShmoys.HasPlanarDrawingOn H := 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).