In this paper, dynamical systems whose structure is defined by means of a simple, directed graph are considered. These objects can be used to model structured systems or, more generally, networks of systems and systems of systems, where the relations between state, input and output variables or, respectively, between agents are known only for being zero or nonzero. Using an approach that is conceptually similar to the geometric approach developed for linear time-invariant systems, suitable notions of invariance, controlled invariance and conditioned invariance are introduced and related to the action of feedbacks. The results are used to provide general solvability conditions for disturbance decoupling problems expressed in graph-theoretic terms.