  We consider gradient systems with an increasing potential that depends on a
scalar parameter. As the parameter is varied, critical points of the potential
can be eliminated or created through saddle-node bifurcations causing the
system to transit from one stable equilibrium located at a (local) minimum
point of the potential to another minimum along the heteroclinic connections.
These transitions can be represented by a graph. We show that any admissible
graph has a realization in the class of two dimensional gradient flows. The
relevance of this result is discussed in the context of genesis of hysteresis
phenomena. The Preisach hysteresis model is considered as an example.
