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We study the ordered configuration spaces of star graphs. Inspired by therepresentation stability results of Church--Ellenberg--Farb for the orderedconfiguration space of a manifold and the edge stability results ofAn--Drummond-Cole--Knudsen for the unordered configuration space of a graph, wedetermine how the ordered configuration space of a star graph with $k$ leavesbehaves as we add particles at the leaves. We show that, as a module over thecombinatorial category FI$_{k, o}$, the first homology of this orderedconfiguration space is finitely generated by $4$ particles for $k=3$, by $3$particles for $k=4$, and by $2$ particles for $k\ge 5$. Additionally, we provethat every relation among homology classes can be described by relations on atmost $6$ particles for $k=4$, at most $5$ particles when $k=5$, at most $4$particles when $k=6$, and at most $3$ particles for $k\ge 7$, while provingthat adding particles always introduces new relations when $k=3$. This provesthat there is no finite universal presentation for the homology of orderedconfiguration spaces of graphs.

Homology Generators and Relations for the Ordered Configuration Space of a Star Graph