Proving that the cover time for graph is exponential in the worst case How can I prove that the cover time for a directed graph $G$ can be exponential in the size of $G$?

The cover time is the expected length of a random walk that visits all vertices.
You do it in two steps:


First you think of a graph which you can expect to be difficult.
Then you prove your suspicion.


Let's start with the first step - do you have any graph in mind?