Abstract In this paper, we consider dynamical systems where the graph of the relations between state, input and output variables switches between different configurations, according to the action of a switching time signal. Moreover, in each configuration the relations between the variables are known only for being zero or nonzero. Switching structured systems of this kind are described by families of simple, directed graphs. They can be used to model complex networks of systems as well as systems of systems for which the only available information consists in the patterns that the set of the interconnections between the components, or agents, may assume in different situations. Using an approach that is conceptually similar to the geometric approach developed for linear time-invariant systems, suitable notions of invariance and controlled invariance are introduced and related to the action of feedback. These notions are used to provide general solvability conditions for the disturbance decoupling problem expressed in graph-theoretic terms.