  The Frechet distance is often used to measure distances between paths, with
applications in areas ranging from map matching to GPS trajectory analysis to
handwriting recognition. More recently, the Frechet distance has been
generalized to a distance between two copies of the same graph embedded or
immersed in a metric space; this more general setting opens up a wide range of
more complex applications in graph analysis. In this paper, we initiate a study
of some of the fundamental topological properties of spaces of paths and of
graphs mapped to R^n under the Frechet distance, in an effort to lay the
theoretical groundwork for understanding how these distances can be used in
practice. In particular, we prove whether or not these spaces, and the metric
balls therein, are path-connected.

Computational Geometry;Geometric Topology