Planar drawing restricts to induced subgraphs
ProvedWilliamsonShmoys.planar_drawing_induce__p2m_8fb060ce_122c_4344_a570_6fdb0412375d_a1approximation-algorithmsplanar-graphstreewidth
If a finite linearly ordered simple graph has the specified crossing-free drawing, then the induced graph on any vertex subset has the same kind of drawing, using restricted vertex points and the original curves.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_induce__p2m_8fb060ce_122c_4344_a570_6fdb0412375d_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) (hN : HasPlanarDrawingOn N) (s : Set V) :
letI : Fintype {v // v ∈ s} := Fintype.ofFinite _
HasPlanarDrawingOn (N.induce s) := 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).