  P-time event graphs (P-TEGs) are specific timed discrete-event systems, in
which the timing of events is constrained by intervals. An important problem is
to check, for all natural numbers $d$, the existence of consistent $d$-periodic
trajectories for a given P-TEG. In graph theory, the
Proportional-Inverse-Constant-Non-positive Circuit weight Problem (PIC-NCP)
consists in finding all the values of a parameter such that a particular
parametric weighted directed graph does not contain circuits with positive
weight. In a related paper, we have proposed a strongly polynomial algorithm
that solves the PIC-NCP in lower worst-case complexity compared to other
algorithms reported in literature. In the present paper, we show that the first
problem can be formulated as an instance of the second; consequently, we prove
that the same algorithm can be used to find $d$-periodic trajectories in
P-TEGs. Moreover, exploiting the connection between the PIC-NCP and max-plus
algebra we prove that, given a P-TEG, the existence of a consistent 1-periodic
trajectory of a certain period is a necessary and sufficient condition for the
existence of a consistent $d$-periodic trajectory of the same period, for any
value of $d$.
