  Multiparameter persistence modules come up naturally in topological data
analysis and topological robotics. Given a metric graph $(X,\delta)$, the
second configuration space of $(X,\delta)$ with proximity parameters (for
example, the minimum distance allowed between each pair of robots) can be
interpreted as a collection of all possible configurations of two robots moving
in $(X,\delta)$. In this project, we study the $2$-parameter persistence
modules associated with the second configuration spaces of the star graph
($\mathsf{Star}_k$, $k\geq 3$) and the generalized H-graph
($\mathcal{H}_{m,n}$, $m,n\geq 3$) with the edge length parameter $L_e$ and the
restraint parameter $r$. Moreover, we provide the indecomposable direct
summands for each persistence module.
