I'm trying to figure out what type of problem this is and whether there exist any efficient algorithm to solve it. Apologies in advance, it's been a while since I wrote in terms of graphs, hopefully, my terminology comes across OK!
Assuming a planar graph $G = (V, E)$, let $P_{s-t}$ denote the set of all cycle free paths between vertices $s$ and $t$ in $G$, where $P_{s-t}^n$ is the $n$-th path in the set. Further, $P_{s-t}^n(i)$ is the $i$-th vertice on the $n$-th path, e.g., $P_{s-t}^n(0) = s$.
My question: Assuming $s, k, t \in V$, does there exist an integer $n$ such that $P^n_{s-t}(i) \neq P^m_{k-t}(i)$ is satisfied for all $m$ and $i$?
Just to clarify, I am looking for an efficient process to verify the above.