Connected planar supergraph when there are at least three vertices
ProvedWilliamsonShmoys.planar_connected_supergraph_of_three_vertices__p2m_b9db41c8_79f8_4b68_86df_fe1252530a28_a3approximation-algorithmsplanar-graphstreewidth
A finite graph with a crossing-free drawing and at least three vertices has a connected simple supergraph on the same vertices that also has a crossing-free drawing. The size hypothesis excludes the empty-type counterexample to the earlier child statement.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_connected_supergraph_of_three_vertices__p2m_b9db41c8_79f8_4b68_86df_fe1252530a28_a3
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hH : HasPlanarDrawingOn H)
(hcard : 3 ≤ Fintype.card V) :
∃ K : SimpleGraph V, H ≤ K ∧ K.Connected ∧ HasPlanarDrawingOn K := 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).