We investigate the relation between the cooperative length and 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, τ_{p}, which is the time until a particle moves for the first time, and the culling time, τ_{c}, which is the minimal number of particles that need to move before a specific particle can move, τ_{p}=τ_{c}^{γ}, where γ 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/τ)^{β}], but unlike previous works, we find that the exponent β appears to decay to 0 as the particle density approaches 1.