  We construct a bounded degree graph $G$, such that a simple random walk on it
is transient but the random walk path (i.e., the subgraph of all the edges the
random walk has crossed) has only finitely many cutpoints, almost surely. We
also prove that the expected number of cutpoints of any transient Markov chain
is infinite. This answers two questions of James, Lyons and Peres [A Transient
Markov Chain With Finitely Many Cutpoints (2007) Festschrift for David
Freedman]. Additionally, we consider a simple random walk on a finite connected
graph $G$ that starts at some fixed vertex $x$ and is stopped when it first
visits some other fixed vertex $y$. We provide a lower bound on the expected
effective resistance between $x$ and $y$ in the path of the walk, giving a
partial answer to a question raised in [Ann. Probab. 35 (2007) 732--738].
