Parity conditions for one-way rail networks

We present parity conditions under which a toy-train rail network is one-way, i.e., every reachable point can be approached from only one direction. 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.