The problem of finding fault patterns consistent with a given syndrome is discussed for graph-theoretical diagnosis models such as the fault-diagnosis and self-diagnosis models. The fault-diagnosis model consists of two types of vertices, fault units and measurements, and is expressed by a bipartite graph. Faulty states of a fault unit always imply abnormal states of all the measurements which are adjacent to the unit, otherwise a measurement remains normal. A self-diagnosis model consists of one type of unit which has the capability of testing other units and being tested itself. The testing relation is represented by a directed arc; this produces test outcomes which are invalid if the testing unit is faulty. The inverse system which yields a fault pattern from a corresponding syndrome for fault-diagnosis models is studied and a syndrome-decoding algorithm is proposed which works for some class of diagnosis models with observation noise. The algorithm uses a similar measure to the syndrome-decoding algorithm of error-correcting codes which use the Hamming distance. Another measure is presented for the self-diagnosis model expressed by a directed graph and this measure is characterized by a ranking method.