  Geometric graphs appear in many real-world data sets, such as road networks,
sensor networks, and molecules. We investigate the notion of distance between
embedded graphs and present a metric to measure the distance between two
geometric graphs via merge trees. In order to preserve as much useful
information as possible from the original data, we introduce a way of rotating
the sublevel set to obtain the merge trees via the idea of the directional
transform. We represent the merge trees using a surjective multi-labeling
scheme and then compute the distance between two representative matrices. We
show some theoretically desirable qualities and present two methods of
computation: approximation via sampling and exact distance using a kinetic data
structure, both in polynomial time. We illustrate its utility by implementing
it on two data sets.
