I can see the case of an aperiodic first cycle length, but given that each vertex in the graph created by the input -> output nodes of the traversal path has only one outgoing edge, is it possible for the cyclic graph from a1 to some z1 to have an inconsistent length? And is there ever going to be some z2 for which the aperiodic cycle length from a1 to either z1 or z2 results in a better answer to the problem given that the periodic cycle length of a1 to z1 is shorter than the cycle length of a1 to z2? Please forgive me if I am not using the correct graph theory terms.