The directed Generalized Geography game is well-known to be PSPACE-complete, however, I could not find anything for the undirected version. I saw that in Hans L. Bodlaender, Complexity of path-forming games, Theoretical Computer Science, Volume 110, Issue 1, 15 March 1993, Pages 215-245 (thx to Marzio De Biasi for the link) it is posed as an open problem. Is it still open?
In fact, I wonder if there is a simple gadget that we can put in the place of every directed edge to make a straight-forward reduction from the directed version. Is it known that no such gadget can exist?