  The isomorphism problem for planar graphs is known to be efficiently
solvable. For planar 3-connected graphs, the isomorphism problem can be solved
by efficient parallel algorithms, it is in the class $AC^1$. In this paper we
improve the upper bound for planar 3-connected graphs to unambiguous logspace,
in fact to $UL \cap coUL$. As a consequence of our method we get that the
isomorphism problem for oriented graphs is in $NL$. We also show that the
problems are hard for $L$.
