  We investigate the relation between the cooperative length and the relaxation
time, represented respectively by the culling time and the persistence time, in
the Fredrickson-Andersen, Kob-Andersen and spiral kinetically-constrained
models. By mapping the dynamics to diffusion of defects, we find a relation
between the persistence time, $\tau_{p}$, which is the time until a particle
moves for the first time, and the culling time, $\tau_{c}$, which is the
minimal number of particles that need to move before a specific particle can
move, $\tau_{p}=\tau^{\gamma}_{c}$, where $\gamma$ is model- and dimension
dependent. We also show that the persistence function in the Kob-Andersen and
Fredrickson-Andersen models decays subexponentially in time,
$P(t)=exp[-(t/\tau)^{\beta}]$, but unlike previous works we find that the
exponent $\beta$ appears to decay to 0 as the particle density approaches 1.
