  We study the decoherence of two coupled spins that interact with a chaotic
spin-bath environment. It is shown that connectivity of spins in the bath is of
crucial importance for the decoherence of the central system. The previously
found phenomenon of two-step decoherence (Phys. Rev. Lett. {\bf 90}, 210401
(2003)) turns out to be typical for the bath with a slow enough dynamics or no
dynamics. For a generic random system with chaotic dynamics a conventional
exponential relaxation to the pointer states takes place. Our results confirm a
conjecture of Paz and Zurek (Phys. Rev. Lett. {\bf 82}, 5181 (1999)) that for
weak enough interactions the pointer states are eigenstates of the central
system.
