We consider discrete-time switched systems with switching of linear time-invariant
right-hand parts. The notion of a connected discrete switched system is introduced. For
systems with the connectedness property, we propose necessary and sufficient frequency-domain
conditions for the existence of a common quadratic Lyapunov function that provides the stability
for a system under arbitrary switching. The set of connected switched systems contains discrete
control systems with several time-varying nonlinearities from the finite sectors, considered in the
theory of absolute stability. We consider the case of switching between three linear subsystems
in more details and give an illustrative example.