Starting BFS at s and t
Yes, this is a known thing. It's called bidirectional search.  It will find shortest paths in unweighted graphs and, in graphs of where every vertex has degree&nbsp;$d$, it will visit at most $2d^{\ell/2}=2\sqrt{d^\ell}$ vertices (where $\ell$&nbsp;is the length of the shortest path), whereas ordinary BFS could visit up to $d^\ell$&nbsp;vertices.