  Trajectories of a signal that fluctuates between two states which originate
from single molecule activities have become ubiquitous. Common examples are
trajectories of ionic flux through individual membrane-channels, and of photon
counts collected from diffusion, activity, and conformational changes of
biopolymers. By analyzing the trajectory, one wishes to deduce the underlying
mechanism, which is usually described by a multi-substate kinetic scheme. In
previous works, we divided kinetic schemes that generate two-state trajectories
into two types: reducible schemes and irreducible schemes. We showed that all
the information in trajectories generated from reducible schemes is contained
in the waiting time probability density functions (PDFs) of the two states. It
follows that reducible schemes with the same waiting time PDFs are not
distinguishable. In this work, we further characterize the topologies of
kinetic schemes, now of irreducible schemes, and further study two-state
trajectories from the two types of scheme. We suggest various methods for
extracting information about the underlying kinetic scheme from the trajectory
(e. g., calculate the binned successive waiting times PDF and analyze the
ordered waiting times trajectory), and point out the advantages and
disadvantages of each. We show that the binned successive waiting times PDF is
not only more robust than other functions when analyzing finite trajectories,
but contains, in most cases, more information about the underlying kinetic
scheme than other functions in the limit of infinitely long trajectories. For
some cases however, analyzing the ordered waiting times trajectory may supply
unique information about the underlying kinetic scheme.
