Extend a nonempty planar graph to a connected planar supergraph
OpenWilliamsonShmoys.planar_drawing_connected_supergraph_exists__p2m_ba26c1d0_6e77_4c7c_aa33_77d1fdc6fec8_a3approximation-algorithmsplanar-graphstreewidth
Every nonempty finite planar graph has a connected planar supergraph on the same vertices.
Preamble
import Mathlib import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_drawing_connected_supergraph_exists__p2m_ba26c1d0_6e77_4c7c_aa33_77d1fdc6fec8_a3
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V)
(hplanar : HasPlanarDrawingOn K)
(hV : Nonempty V) :
∃ L : SimpleGraph V,
K.edgeSet ⊆ L.edgeSet ∧
L.Connected ∧
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).