a line segment and $|V| = 2$, then $\pi_1(\operatorname{Conf}_n^{\operatorname{sink}}(G,V))$ is free, whereas if $|V| = 3$ then $\pi_1(\operatorname{Conf}_n^{\operatorname{sink}}(G,V))$ is right-angled Artin.\

This theorem was proven in the case wherein $G$ is a line segment with two vertices in [@CL]. Note that Theorem \[treepi1\] is proven via an explicit induction argument and Van Kampen’s Theorem. It will follow that one may always, in principle, compute these fundamental groups explicitly (see Theorem \[treepi\]). At this time it is unclear whether $\pi_1(\operatorname{Conf}_n^{\operatorname{sink}}(G,V))$ is always right-angled Artin, as it is unclear whether one can choose a presentation wherein the relations are commutators of generators, as opposed to just commutators. This theorem is also significant, as it is the first evidence of the following fact: the spaces $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ will, in general, depend heavily on the vertex and edge sets of $G$. In particular, $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ is not a homeomorphism invariant of the graph $G$, although it is invariant under cellular homeomorphisms. This is in sharp contrast to the classical configuration spaces of graphs, which are unchanged by introducing vertices to $G$.\
While the cellular model of Theorem \[fcm\] is convenient for developing visual intuition for the sink configuration spaces (see Figure \[cellexample\]), it is not quite 