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 ) space, for any 0"> 0 . The previous best known space bound for this problem with polynomial time was approximately O ( 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.