A special class of hybrid systems consisting of switched linear systems is considered. A subclass of such systems is emphasized for which necessary and sufficient existence conditions (a criterion) of a common quadratic Lyapunov function are obtained as sufficient stability conditions for that switched system. Systems of this subclass are called connected switched linear systems. This subclass includes control systems with some sector constrained time-varying nonlinearities considered in the absolute stability theory. For connected switched linear systems of a particular type that is called triangular, there established either necessary or sufficient existence conditions of such Lyapunov functions. The relation between these conditions is considered in an example presented.