  We present an alternative application of discrete Morse theory for
two-particle graph configuration spaces. In contrast to previous constructions,
which are based on discrete Morse vector fields, our approach is through Morse
functions, which have a nice physical interpretation as two-body potentials
constructed from one-body potentials. We also give a brief introduction to
discrete Morse theory. Our motivation comes from the problem of quantum
statistics for particles on networks, for which generalized versions of anyon
statistics can appear.
