What time complexity is a reachability algorithm?
You can calculate all reachable nodes in $O(V+E)$ time and $O(V)$ space, using DFS on the fly.

If you want to use less than $O(V)$ space, then the problem becomes much more challenging.  I recommend looking at how the SPIN model checker works.  See https://en.wikipedia.org/wiki/Bitstate_hashing.  Even then, the space required is probably still $O(V)$, just with a lower constant factor.