Drawing Graphs as Spanners

We study the problem of embedding graphs in plane as good geometric spanners. That is, for a graph G, goal is to construct straight-line drawing $$\Gamma $$ G such that, any two vertices u and v ratio between minimum length path from Euclidean distance small. The maximum ratio, over all pairs spanning . First, we show that deciding whether admits with 1, proper planar 1 are NP-complete, $$\exists \mathbb R$$ -complete, linear-time solvable problems, respectively, where if no overlap edge overlaps vertex. Second, moving $$1+\epsilon allows us draw every graph. Namely, prove $$\epsilon >0$$ , (planar) (resp. planar) smaller than Third, our drawings have large edge-length longest shortest exponential. this sometimes unavoidable. More generally, identify having bounded toughness criterion distinguishes admit constant polynomial require exponential ratio.