Metric Spaces of Graphs
It is often useful to study spaces of graphs living in a metric ambient space for many application areas in GIS and networks. Some of my research studies and defines these spaces rigorously.
Topological Properties of the Fréchet distance on Graphs
Preprint awaiting submission. Any feedback to improve this manuscript is encouraged!
Title: Metric and Topological Properties of Paths and Graphs Under the Fréchet Distance
Abstract:
The Fréchet distance is often used to measure distances between paths, with applications arising in road network and GPS trajectory analysis. The Fréchet distance can also be used to study distances between two copies of the same graph embedded or immersed in a metric space. In this paper, we examine topological properties of spaces of paths and of graphs mapped to Rn under the Fréchet distance. In particular, we prove whether or not these spaces and metric balls in these spaces are path- connected.
YRF 2022
Presented in the Young Researchers Forum accompanying the 2022 CG Week in Berlin.
Title: ‘Path-Connectivity of Fr'echet Spaces of Graphs’
Abstract:
We examine topological properties of spaces of paths and graphs mapped to $R^d$ under the Fréchet distance. We show that the spaces of graphs and paths mapped to $R^d$ are path-connected if the map is either continuous or an immersion. If the map is an embedding, we show that the space of paths is path-connected, while the space of graphs only maintains this property in dimension 4 or higher.