Planarity is inherited by spanning subgraphs
ProvedWilliamsonShmoys.planarDrawingOn_mono__p2m_19cd2d46_2b88_4ae0_bdf6_b262bd78d159_a1approximation-algorithmsplanar-graphstreewidth
A crossing-free drawing of a graph also witnesses a crossing-free drawing of any spanning subgraph, since the latter has only edges already represented in the drawing.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planarDrawingOn_mono__p2m_19cd2d46_2b88_4ae0_bdf6_b262bd78d159_a1
{V : Type} [Fintype V] [LinearOrder V]
(G H : SimpleGraph V)
(hG : WilliamsonShmoys.HasPlanarDrawingOn G)
(hsub : ∀ a b, H.Adj a b → G.Adj a b) :
WilliamsonShmoys.HasPlanarDrawingOn H := 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).