Parity conditions for one-way rail networks

We present parity conditions under which a toy rail network is one-way, i.e., whether a direction can be assigned across the network so that all train journeys are completely consistent with it or completely consistent with its opposite. We show that this problem is equivalent to determining the balance of a signed graph obtained from the network, whose edges are assigned positive or negative signs. Using signed-graph theory, we derive two equivalent parity conditions for one-wayness: (i) every cycle must contain an even number of edges that join the same sides of switches, and (ii) every cycle must contain an even number of angles at switches. Signed-graph theory also offers an analytical criterion: A connected network is one-way if and only if the smallest eigenvalue of its signed Laplacian matrix is zero, suggesting a computational tool for evaluating one-wayness.