We study the problem of space-efficient

polynomial-time algorithms for {\em directed

st-connectivity} (STCON).

Given a directed graph $G$, and a pair of vertices $s, t$, the STCON problem

is to decide if there exists a path from $s$ to $t$ in $G$.

For general graphs, the best polynomial-time algorithm for STCON

uses space that is only slightly sublinear.

However, for special classes of directed graphs, polynomial-time poly-logarithmic-space

algorithms are known for STCON. In this paper, we continue this thread of research

and study a class of graphs called

\emph{unique-path graphs with respect to source $s$},

where there is at most one simple path from $s$ to any vertex in the graph.

For these graphs, we give

a polynomial-time algorithm that uses

$\tilde O(n^{\varepsilon})$ space for any constant $\varepsilon \in (0,1]$.

We also give a polynomial-time, $\tilde O(n^\varepsilon)$-space

algorithm to \emph{recognize} unique-path graphs.

Unique-path graphs are related to configuration graphs of unambiguous

log-space computations, but they can have some directed cycles. Our results

may be viewed along the continuum of sublinear-space polynomial-time

algorithms for STCON in different classes of directed graphs - from

slightly sublinear-space algorithms for general graphs to $O(\log n)$ space algorithms for trees.