  We report progress on the \NL vs \UL problem. [-] We show unconditionally
that the complexity class $\ReachFewL\subseteq\UL$. This improves on the
earlier known upper bound $\ReachFewL \subseteq \FewL$. [-] We investigate the
complexity of min-uniqueness - a central notion in studying the \NL vs \UL
problem. We show that min-uniqueness is necessary and sufficient for showing
$\NL =\UL$. We revisit the class $\OptL[\log n]$ and show that {\sc
ShortestPathLength} - computing the length of the shortest path in a DAG, is
complete for $\OptL[\log n]$. We introduce $\UOptL[\log n]$, an unambiguous
version of $\OptL[\log n]$, and show that (a) $\NL =\UL$ if and only if
$\OptL[\log n] = \UOptL[\log n]$, (b) $\LogFew \leq \UOptL[\log n] \leq \SPL$.
[-] We show that the reachability problem over graphs embedded on 3 pages is
complete for \NL. This contrasts with the reachability problem over graphs
embedded on 2 pages which is logspace equivalent to the reachability problem in
planar graphs and hence is in \UL.
