  A metric graph is a geometric realization of a finite graph by identifying
each edge with a real interval. A divisor on a metric graph $\Gamma$ is an
element of the free abelian group on $\Gamma$. The rank of a divisor on a
metric graph is a concept appearing in the Riemann-Roch theorem for metric
graphs (or tropical curves) due to Gathmann and Kerber, and Mikhalkin and
Zharkov. We define a \emph{rank-determining set} of a metric graph $\Gamma$ to
be a subset $A$ of $\Gamma$ such that the rank of a divisor $D$ on $\Gamma$ is
always equal to the rank of $D$ restricted on $A$. We show constructively in
this paper that there exist finite rank-determining sets. In addition, we
investigate the properties of rank-determining sets in general and formulate a
criterion for rank-determining sets. Our analysis is a based on an algorithm to
derive the $v_0$-reduced divisor from any effective divisor in the same linear
system.
