A disconnected drawable graph with enough vertices has a drawable extension joining distinct components
OpenWilliamsonShmoys.drawable_extension_by_joining_components__p2m_10785777_128a_4e88_8513_e417451ea141_a2approximation-algorithmsplanar-graphstreewidth
Every disconnected drawable graph on at least three vertices has a drawable proper supergraph formed by adding an edge between vertices that were in different components.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.drawable_extension_by_joining_components__p2m_10785777_128a_4e88_8513_e417451ea141_a2
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V) (hM : HasPlanarDrawingOn M)
(hcard : 3 ≤ Fintype.card V)
(hdisc : ¬ M.Connected) :
∃ u v : V, ¬ M.Reachable u v ∧
∃ N : SimpleGraph V, M ≤ N ∧ N.Adj u v ∧
HasPlanarDrawingOn N ∧ N ≠ M := 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).