This paper analyses discrete-time switched systems in which a control engineer designs each mode so that performance guarantees are met. Given this setting, our objective is then to identify the set of mode transitions so that stability is guaranteed under arbitrary switching. Namely, our specific objective is to identify the stability graph associated with a given switching system. These systems are important for multi-controller architectures in many practical applications. Specifically, we develop convex conditions that ensure asymptotic stability and guaranteed quadratic performance for constrained switched systems. Then, a greedy algorithm based on the graph associated with the switched system is designed in order to determine a set of switching functions for which stability is guaranteed. We then further restrict the graph so that performance metrics are met. Simulations are given to illustrate the efficacy of the procedures.