Rank-determining sets of metric graphs

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 $Γ$ is an element of the free abelian group on $Γ$. 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 $Γ$ to be a subset $A$ of $Γ$ such that the rank of a divisor $D$ on $Γ$ 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.

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