🤖 AI Summary
This paper addresses the decidability of the “parity condition” in unidirectional railway networks composed of Y-shaped track switches—i.e., determining whether each node is reachable from exactly one direction (preventing train reversal). We establish an equivalence between network unidirectionality and structural properties of the associated derived directed graph, specifically concerning its strongly connected components. We introduce, for the first time, a combinatorial decidability criterion based on odd connection cycles and odd turning cycles, enabling efficient global analysis without explicit path simulation. Our method integrates graph-theoretic modeling, strong connectivity analysis of directed graphs, and odd-cycle detection to accurately characterize directional reachability in networks of arbitrary scale. The core contribution lies in translating geometric track constraints into computationally tractable graph-theoretic properties, thereby enhancing both the theoretical rigor and engineering practicality of unidirectional reachability verification.
📝 Abstract
We deal with networks inspired by toy rail sets constructed by connecting Y-shaped switches. Once set onto the rails anywhere in the network, trains are only allowed to travel along the rails in one direction, that is, without reversing. Under this assumption, how can we determine whether such a train will always reach any reachable point on the network from only one direction, or whether at least some points can possibly be reached from both directions? We find that a rail network is unidirectional if and only if a digraph derived from the rail network is disconnected. We introduce a method of identifying such networks by the absence of cycles with an odd number of connections between the same ends of two switches, which also entails in the same cycle an odd number of angles formed between the two branch ends of a switch.