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The directed ST-connectivity problem asks whether a directed graph contains a graph path from specified vertices $s$ to $t$. A zero-length graph path is permitted, so $s=t$ is a positive instance. It is NL-complete under logspace many-one reductions.

For membership in NL, guess a graph path of at most $N-1$ edges, where $N$ is the number of vertices. Retain only the current vertex and a step counter. A reachable target has a simple graph path within that bound, and the stored indices use logarithmic space.

For hardness, take an NL machine on a fixed input and construct its configuration graph. Only polynomially many configurations are possible within the workspace bound, and all configurations and local transition edges can be enumerated with a logspace transducer. Add one target collecting the accepting configurations. The target is reachable from the initial configuration exactly when the original machine accepts. This translates any NL language into directed reachability while respecting the reduction's storage bound.