Connecting the components of a crossing-free drawing
ProvedWilliamsonShmoys.planar_drawing_connected_supergraph__p2m_ef56db37_3285_4935_af76_5046b2776a5c_a1approximation-algorithmsplanar-graphstreewidth
A graph on at least three finite ordered vertices with a HasPlanarDrawingOn drawing is a subgraph of a connected graph on the same vertices that also has a HasPlanarDrawingOn drawing.
Preamble
import Mathlib import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_connected_supergraph__p2m_ef56db37_3285_4935_af76_5046b2776a5c_a1 {V : Type} [Fintype V] [LinearOrder V] (K : SimpleGraph V) (hcard : 3 <= Fintype.card V) (hK : WilliamsonShmoys.HasPlanarDrawingOn K) : ∃ L : SimpleGraph V, K ≤ L ∧ L.Connected ∧ 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).