Fault detection and isolation (FDI) problems are here considered for linear systems with faults and disturbances. We design a set of observer-based residuals in such a way that the transfer from the disturbances to the residuals is zero, and the transfer from the faults to the residuals allows fault isolation. We are interested in obtaining a transfer function from faults to residuals with either a diagonal structure (Le, a dedicated structured residuals set) or a triangular one. We deal with this problem when the system under consideration is structured. That is, the entries of the system matrices are either fixed zeros or free parameters. To a structured system, one am associate in a natural way a directed graph. We can then provide necessary and sufficient conditions under which the FDI problems have a solution for almost any value of the free parameters. These conditions are simply expressed in terms of input-output paths in the associated graph.