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