A disconnected drawable graph has a strict drawable extension
OpenWilliamsonShmoys.disconnected_planar_graph_has_drawable_extension__p2m_eec88a46_e2f1_426a_bbcb_0322b031599d_a1approximation-algorithmsplanar-graphstreewidth
A disconnected graph on a finite linearly ordered vertex type with at least three vertices, if it has a crossing-free drawing, has a strict supergraph on the same vertices that also has a crossing-free drawing.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.disconnected_planar_graph_has_drawable_extension__p2m_eec88a46_e2f1_426a_bbcb_0322b031599d_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hN : HasPlanarDrawingOn N) (hdisconn : ¬ N.Connected) :
∃ P : SimpleGraph V, N ≤ P ∧ P ≠ N ∧ 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).