  We consider the graph dynamical systems known as k-reversible processes. In
such processes, each vertex in the graph has one of two possible states at each
discrete time step. Each vertex changes its state between the current time and
the next if and only if it currently has at least k neighbors in a state
different than its own. For such processes, we present a monotonic function
similar to the decreasing energy functions used to study threshold networks.
Using this new function, we show an alternative proof for the maximum period
length in a k-reversible process and provide better upper bounds on the
transient length in both the general case and the case of trees.
