Restrict a planar drawing to an induced subgraph
ProvedWilliamsonShmoys.planar_drawing_induce__p2m_c4d63dfe_c88f_43bb_a642_10d75b5a8453_a2approximation-algorithmsplanar-graphstreewidth
Restricting a planar drawing to an induced vertex subset gives a drawing satisfying the same conditions for the induced subgraph.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_induce__p2m_c4d63dfe_c88f_43bb_a642_10d75b5a8453_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (S : Finset V)
(hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
WilliamsonShmoys.HasPlanarDrawingOn (K.induce (↑S : Set V)) := 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).