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

Homeomorphism and Homotopy Types of Restricted Configuration Spaces of Metric Graphs