Partition Graphs of Metric Spaces

Partitioning a compact metric space and building an associated graph allows us to show the underlying structure and relationships within the space. In this paper, we aim to explore the fundamental notions of graphs and trees in order to build the necessary understanding for creating partition graphs. We will then show how the ends of the partition graphs created in such a manner are not only homeomorphic to the original compact metric space but also bi-H ¨older to it.