# Rank-determining sets of metric graphs - Mathematics > Combinatorics

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Abstract: A metric graph is a geometric realization of a finite graph by identifyingeach edge with a real interval. A divisor on a metric graph $\Gamma$ is anelement of the free abelian group on $\Gamma$. The rank of a divisor on ametric graph is a concept appearing in the Riemann-Roch theorem for metricgraphs or tropical curves due to Gathmann and Kerber, and Mikhalkin andZharkov. We define a \emph{rank-determining set} of a metric graph $\Gamma$ tobe a subset $A$ of $\Gamma$ such that the rank of a divisor $D$ on $\Gamma$ isalways equal to the rank of $D$ restricted on $A$. We show constructively inthis paper that there exist finite rank-determining sets. In addition, weinvestigate the properties of rank-determining sets in general and formulate acriterion for rank-determining sets. Our analysis is a based on an algorithm toderive the $v 0$-reduced divisor from any effective divisor in the same linearsystem.

Author: ** Ye Luo**

Source: https://arxiv.org/