Extend a planar drawing across distinct components
OpenWilliamsonShmoys.planar_add_edge_between_nonreachable_vertices_geometric_extension__p2m_61c492cb_f898_4cbe_b4f0_23e57d6c2689_a3approximation-algorithmsplanar-graphstreewidth
If a finite graph with a drawing has two vertices in different connected components, there is a drawable supergraph containing the edge between those vertices; that edge was absent from the original graph.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_add_edge_between_nonreachable_vertices_geometric_extension__p2m_61c492cb_f898_4cbe_b4f0_23e57d6c2689_a3
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) (u v : V)
(hsep : ¬ N.Reachable u v) (hN : HasPlanarDrawingOn N) :
∃ P : SimpleGraph V, N ≤ P ∧ P.Adj u v ∧ ¬ N.Adj u v ∧ HasPlanarDrawingOn P := 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).