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

Metric and Path-Connectedness Properties of the Frechet Distance for Paths and Graphs