A sufficiently large disconnected drawable graph has a drawable strict supergraph
OpenWilliamsonShmoys.disconnected_drawable_extension__p2m_34e3c7ce_8936_45db_894d_7bc025c10cb3_a2approximation-algorithmsplanar-graphstreewidth
A drawable graph on at least three vertices that is disconnected has a drawable strict supergraph on the same vertices. This isolates the edge-addition geometry from the parent; the size condition prevents the empty-vertex counterexample to the earlier version. A proof can redraw the finite components in separated regions and route an arc between two components through the complement, while preserving all conditions in HasPlanarDrawingOn. Formalizing that construction is the remaining core obligation.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.disconnected_drawable_extension__p2m_34e3c7ce_8936_45db_894d_7bc025c10cb3_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K)
(hcard : 3 <= Fintype.card V)
(hDisconnected : ¬ K.Connected) :
∃ L : SimpleGraph V, K ≤ L ∧ L ≠ K ∧ WilliamsonShmoys.HasPlanarDrawingOn L := 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).