  Given a graph $G$ and two vertices $s$ and $t$ in it, {\em graph
reachability} is the problem of checking whether there exists a path from $s$
to $t$ in $G$. We show that reachability in directed layered planar graphs can
be decided in polynomial time and $O(n^\epsilon)$ space, for any $\epsilon >
0$. The previous best known space bound for this problem with polynomial time
was approximately $O(\sqrt{n})$ space \cite{INPVW13}.
  Deciding graph reachability in {\SC} is an important open question in
complexity theory and in this paper we make progress towards resolving this
question.
