System level diagnosis has been studied for many years since distributed systems became available. Preparata, et al. introduced the PMC model for identifying faulty units in a system using a graph, and at the same time gave the concept of
t-diagnosability. Various measures of fault diagnosability have been proposed since then. In this paper, we provide characterization theorems for two measures of fault identification capability of a system which were introduced by Somani and Amin. We consider a structure of the system which has large capability for the other measures. In Somani's model for (
k,
t)-diagnosability, where a
t-diagnosable system suffers at most
k faults at each diagnosis step, we present a polynomial algorithm for obtaining the maximum
t in order for the system to be (
k,
t)-diagnosable given
k. In Amin's model for
r-weak diagnosability, where a
t-diagnosable system has at most one set of
r faulty units tested by at least one normal unit, we examine a structure of the system to have the maximum
r.
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