1-dimensional CW-complex G, two non-isomorphic cellular models of $G$ can yield sink configuration spaces with drastically different topologies.\

Let $(G,V,E)$ be a graph. The is the topological space $$\operatorname{Conf}_n(G) = \{(x_1,\ldots,x_n) \in G^n \mid x_i \neq x_j \text{ if $i \neq j$}\}.$$\

Configuration spaces of graphs have seen a surge of interest due to their connections with robotics [@G][@Far], as well as their many interesting theoretical properties (see [@KP][@A][@FS][@BF][@CL], for a sampling). There has also been a recent push in the literature to understand these spaces from the perspective of representation stability [@L1][@L2][@R1]. In this paper we will use the existing literature on the configuration spaces of graphs as a guiding tool for developing a theoretical framework for studying what we call sink configuration spaces.\

\[sinkdef\] Let $(G,V,E)$ be a graph. The **$n$-th sink configuration space of $(G,V,E)$** is the topological space $$\operatorname{Conf}_n^{\operatorname{sink}}(G,V) = \{(x_1,\ldots,x_n) \in G^n \mid x_i \neq x_j \text{ or } x_i = x_j \in V\}.$$\

The cases $([0,1],\{0,1\},\{(0,1)\})$ and $(S^1,\{0\},\{(0,1)\})$ were studied in [@CL]. The work [@CL] is, to the knowledge of the author, also the first paper where these spaces were considered. Note that, unlike with normal configuration spaces, $\operatorname{Conf}^{\operatorname{sink}}_n(G,V)$ depends on the choice of 