so, we consider spaces where points are only allowed to overlap on vertices of the graph. More formally, let $G$ be a graph with vertex set $V$. Then we consider the spaces, $$\operatorname{Conf}_n^{\operatorname{sink}}(G,V) := \{(x_1,\ldots,x_n) \in G^n \mid x_i \neq x_j \text{ or } x_i = x_j \in V\}.$$ We refer to $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ as the **$n$-strand sink configuration space of $G$**. These spaces were first considered by Chettih and Lütgehetmann in [@CL], although they only look at the cases of a circle, and a line segment. In this work, our goal will be to develop the theory of these spaces for all graphs.\

Topological aspects of $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$
-------------------------------------------------------------------------

We will begin our study of sink configuration spaces from a more traditional topological perspective. In particular, we will initially spend time constructing a cellular model for the spaces $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$.\

\[fcm\] Let $G$ be a graph with vertex set $V$ and edge set $E$. Then the sink configuration space $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ is homotopy equivalent to a CW complex of dimension at most $|E|$. If $G$ is a tree, then this CW complex is a cubical complex.\

In the case of classical configuration spaces of graphs, Abrams has shown [@A] that $\operatorname{Conf}_n(G)$ is homotopy 