  If $G$ is a graph with vertex set $V$, let Conf$_n^{\text{sink}}(G,V)$ be the
space of $n$-tuples of points on $G$, which are only allowed to overlap on
elements of $V$. We think of Conf$_n^{\text{sink}}(G,V)$ as a configuration
space of points on $G$, where points are allowed to collide on vertices. In
this paper, we attempt to understand these spaces from two separate, but
closely related, perspectives. Using techniques of combinatorial topology we
compute the fundamental groups and homology groups of
Conf$_n^{\text{sink}}(G,V)$ in the case where $G$ is a tree. Next, we use
techniques of asymptotic algebra to prove statements about
Conf$_n^{\text{sink}}(G,V)$, for general graphs $G$, whenever $n$ is
sufficiently large. It is proven that, for general graphs, the homology groups
exhibit generalized representation stability in the sense of previous work of
the author.

Algebraic Topology;Combinatorics;Representation Theory