  DNA rearrangement processes recombine gene segments that are organized on the
chromosome in a variety of ways. The segments can overlap, interleave or one
may be a subsegment of another. We use directed graphs to represent segment
organizations on a given locus where contigs containing rearranged segments
represent vertices and the edges correspond to the segment relationships. Using
graph properties we associate a point in a higher dimensional Euclidean space
to each graph such that cluster formations and analysis can be performed with
methods from topological data analysis. The method is applied to a recently
sequenced model organism \textit{Oxytricha trifallax}, a species of ciliate
with highly scrambled genome that undergoes massive rearrangement process after
conjugation. The analysis shows some emerging star-like graph structures
indicating that segments of a single gene can interleave, or even contain all
of the segments from fifteen or more other genes in between its segments. We
also observe that as many as six genes can have their segments mutually
interleaving or overlapping.
