 Even-length directed reachability

Decide whether a finite directed graph has an even-length directed walk from a specified source $s$ to a specified target $t$. Vertices may repeat and a zero-length walk is allowed. The NL-completeness proof uses the graph of pairs $(v,b)$ with $b\in \{0,1\}$ tracking integer parity: every original edge flips $b$, and acceptance means reaching $(t,0)$ from $(s,0)$. At most $2|V|-1$ steps are needed in the parity graph. For hardness, subdivide every edge of an ordinary reachability instance into two directed edges. Both constructions are computable by logspace transducers. Requiring a simple path in a graph in the original graph changes the question; deleting an odd closed portion need not preserve integer parity.