manifold of dimension at least 2. More recently, there have been various efforts made to understand configuration spaces of graphs (see [@G][@A][@KP][@BF][@CL], for instance). In this context, a **graph** is a connected, compact, CW-complex of dimension 1. If $Y$ is a graph with at least one vertex of degree $\geq 3$, then it is a fact that the spaces $\operatorname{Conf}_n(Y)$ are always connected [@A]. In spite of this, there is a sense in which it is “harder” for points on a graph to move around each other than it would be in the case of higher dimensional manifolds. This fact manifests itself in many ways in the study of these spaces, and makes certain aspects of the theory much harder than the analogous aspects in the configuration spaces of manifolds.\
One approach to mitigating this issue is by allowing for certain collisions. For instance, in [@Che] Chettih considers configuration spaces wherein you only disallow more than $k$ coordinates from coinciding, for some $k$. In this paper, we also consider configuration spaces of graphs with certain permitted collisions. Unlike the work of Chettih, where points are allowed to collide anywhere on $G$ so long as less than $k$ points are doing 