  We revisit the parameterized model checking problem for token-passing systems
and specifications in indexed $\textsf{CTL}^\ast \backslash \textsf{X}$.
Emerson and Namjoshi (1995, 2003) have shown that parameterized model checking
of indexed $\textsf{CTL}^\ast \backslash \textsf{X}$ in uni-directional token
rings can be reduced to checking rings up to some \emph{cutoff} size. Clarke et
al. (2004) have shown a similar result for general topologies and indexed
$\textsf{LTL} \backslash \textsf{X}$, provided processes cannot choose the
directions for sending or receiving the token.
  We unify and substantially extend these results by systematically exploring
fragments of indexed $\textsf{CTL}^\ast \backslash \textsf{X}$ with respect to
general topologies. For each fragment we establish whether a cutoff exists, and
for some concrete topologies, such as rings, cliques and stars, we infer small
cutoffs. Finally, we show that the problem becomes undecidable, and thus no
cutoffs exist, if processes are allowed to choose the directions in which they
send or from which they receive the token.
