Connect a planar drawing with at least three vertices
OpenWilliamsonShmoys.planar_drawing_connected_extension_card_ge_three__p2m_ad7657d2_129c_4ff8_bf1d_16d96be983d7_a2approximation-algorithmsplanar-graphstreewidth
A finite graph with a crossing-free drawing and at least three vertices has a connected planar supergraph on the same vertex type.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_drawing_connected_extension_card_ge_three__p2m_ad7657d2_129c_4ff8_bf1d_16d96be983d7_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) [DecidableRel H.Adj]
(hH : HasPlanarDrawingOn H)
(hcard : 3 ≤ Fintype.card V) :
∃ G : SimpleGraph V, H ≤ G ∧ G.Connected ∧ HasPlanarDrawingOn G := 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).