Homeomorphism and Homotopy Types of Restricted Configuration Spaces of Metric Graphs

For Gamma a finite, connected metric graph, we consider the space of configurations of n points in Gamma with a restraint parameter r dictating the minimum distance allowed between each pair of points. These restricted configuration spaces come up naturally in topological robotics. In this paper, we study the homotopy, homeomorphism, and isotopy types of these spaces over the space of parameters r and provide a polynomial upper bound (in the number of edges of Gamma) for the number of isotopy types.