Assemble a connected drawable extension from bridge steps
DisprovedWilliamsonShmoys.planar_drawing_iterated_bridge_connected__p2m_02da0ff6_1020_4ffd_8f79_c5b3eb2fc9a4_a2approximation-algorithmsplanar-graphstreewidth
If every drawable supergraph of a finite graph can be extended by a drawable step connecting any selected unreachable pair, then there is a connected drawable supergraph. Choose an edge-maximal drawable supergraph from the finite set of graphs on V; a disconnected maximal choice would admit a strict bridge extension.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_drawing_iterated_bridge_connected__p2m_02da0ff6_1020_4ffd_8f79_c5b3eb2fc9a4_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) [DecidableRel H.Adj]
(hH : HasPlanarDrawingOn H)
(hStep : ∀ (K : SimpleGraph V), H ≤ K → HasPlanarDrawingOn K →
∀ (a b : V), ¬ K.Reachable a b →
∃ L : SimpleGraph V, K ≤ L ∧ HasPlanarDrawingOn L ∧ L.Reachable a b) :
∃ G : SimpleGraph V, H ≤ G ∧ HasPlanarDrawingOn G ∧ G.Connected := 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).