Planar drawings restrict to subgraphs
ProvedWilliamsonShmoys.hasPlanarDrawingOn_mono__p2m_e2fe7145_1332_437c_98e8_1da8e1e565cf_a1approximation-algorithmsplanar-graphstreewidth
If one graph's edge set is contained in another's, any drawing satisfying the given predicate for the larger graph also satisfies it for the subgraph.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.hasPlanarDrawingOn_mono__p2m_e2fe7145_1332_437c_98e8_1da8e1e565cf_a1
{V : Type} [Fintype V] [LinearOrder V]
{G H : SimpleGraph V} (hGH : G.edgeSet ⊆ H.edgeSet)
(hH : HasPlanarDrawingOn H) : HasPlanarDrawingOn G := 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).