  We study space and time efficient quantum algorithms for two graph problems
-- deciding whether an $n$-vertex graph is a forest, and whether it is
bipartite. Via a reduction to the s-t connectivity problem, we describe quantum
algorithms for deciding both properties in $\tilde{O}(n^{3/2})$ time and using
$O(\log n)$ classical and quantum bits of storage in the adjacency matrix
model. We then present quantum algorithms for deciding the two properties in
the adjacency array model, which run in time $\tilde{O}(n\sqrt{d_m})$ and also
require $O(\log n)$ space, where $d_m$ is the maximum degree of any vertex in
the input graph.
