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

We study the ordered configuration spaces of star graphs. Inspired by the representation stability results of Church--Ellenberg--Farb for the ordered configuration space of a manifold and the edge stability results of An--Drummond-Cole--Knudsen for the unordered configuration space of a graph, we determine how the ordered configuration space of a star graph with $k$ leaves behaves as we add particles at the leaves. We show that, as a module over the combinatorial category FI$_{k, o}$, the first homology of this ordered configuration 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 prove that every relation among homology classes can be described by relations on at most $6$ 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 proving that adding particles always introduces new relations when $k=3$. This proves that there is no finite universal presentation for the homology of ordered configuration spaces of graphs.