The question is essentially in the title: while attempting to compute the colimit of a diagram of cell complexes, my colleague and I find ourselves stumped with the following graph theoretic problem:
Given a directed (unweighted) graph $\Gamma$ and vertices $x,y$, does there exist a vertex $z$ along with directed paths $\gamma:x \to z$ and $\eta:y \to z$ so that the lengths of $\gamma$ and $\eta$ are equal?
The annoying aspect is that these paths need not be the shortest paths connecting $x$ or $y$ to $z$, so everything I can think of (or easily find in a textbook) goes out of the window. For instance, the usual Dijkstra's algorithm and breadth-first search type approaches don't (I think!) yield anything useful when applied naively.
My question is
Does there exist an efficient algorithm to solve this equidistant vertex problem? If not, is there any algorithm which performs better than the dumb path-enumeration?
Thanks for your help!