  Graphs drawn in the plane are ubiquitous, arising from data sets through a
variety of methods ranging from GIS analysis to image classification to shape
analysis. A fundamental problem in this type of data is comparison: given a set
of such graphs, can we rank how similar they are, in such a way that we capture
their geometric "shape" in the plane? In this paper we explore a method to
compare two such embedded graphs, via a simplified combinatorial representation
called a tail-less merge tree which encodes the structure based on a fixed
direction. First, we examine the properties of a distance designed to compare
merge trees called the branching distance, and show that the distance as
defined in previous work fails to satisfy some of the requirements of a metric.
We incorporate this into a new distance function called average branching
distance to compare graphs by looking at the branching distance for merge trees
defined over many directions. Despite the theoretical issues, we show that the
definition is still quite useful in practice by using our open-source code to
cluster data sets of embedded graphs.
