  In recent years, considerable advances have been made in the study of
properties of metric spaces in terms of their doubling dimension. This line of
research has not only enhanced our understanding of finite metrics, but has
also resulted in many algorithmic applications. However, we still do not
understand the interaction between various graph-theoretic (topological)
properties of graphs, and the doubling (geometric) properties of the
shortest-path metrics induced by them. For instance, the following natural
question suggests itself: \emph{given a finite doubling metric $(V,d)$, is
there always an \underline{unweighted} graph $(V',E')$ with $V\subseteq V'$
such that the shortest path metric $d'$ on $V'$ is still doubling, and which
agrees with $d$ on $V$.} This is often useful, given that unweighted graphs are
often easier to reason about.
  We show that for any metric space $(V,d)$, there is an \emph{unweighted}
graph $(V',E')$ with shortest-path metric $d':V'\times V' \to \R_{\geq 0}$ such
that
  -- for all $x,y \in V$, the distances $d(x,y) \leq d'(x,y) \leq (1+\eps)
\cdot d(x,y)$, and
  -- the doubling dimension for $d'$ is not much more than that of $d$, where
this change depends only on $\e$ and not on the size of the graph.
  We show a similar result when both $(V,d)$ and $(V',E')$ are restricted to be
trees: this gives a simpler proof that doubling trees embed into constant
dimensional Euclidean space with constant distortion. We also show that our
results are tight in terms of the tradeoff between distortion and dimension
blowup.
