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 the intersection of UL and co-UL. As a consequence of our method we get that the isomorphism problem for oriented graphs is in NL. We also show that these problems are hard for logspace.