We view a distributed system as a graph of active locations with unidirectional channels between them, through which they pass messages. In this context, the graph structure of a system constrains the propagation of information through it. Suppose a set of channels is a cut set between an information source and a potential sink. We prove that, if there is no disclosure from the source to the cut set, then there can be no disclosure to the sink. We introduce a new formalization of partial disclosure, called *blur operators*, and show that the same cut property is preserved for disclosure to within a blur operator. This cut-blur property also implies a compositional principle, which ensures limited disclosure for a class of systems that differ only beyond the cut.