Abstract The diagnosability of a multiprocessor system is of great significance in measuring the reliability and faulty tolerance of interconnection networks. In this paper, we firstly study the diagnosability of a class of Cayley graphs $Cay(H_n,S_n)$ under the PMC model. We prove that $Cay(H_n,S_n)-F$ keeps the strong local diagnosability property even if it has the set $F$ of $(m-2)$ faulty edges and $m-2$ is maximum number of faulty edges, where $m$ is the regular degree of $Cay(H_n,S_n)$. Secondly, we study the diagnosability of $Cay(H_n,S_n)$ with conditional faulty edges under the PMC model. We prove that $Cay(H_n,S_n)-F$ keeps strong local diagnosability property even if it has the set $F$ of $(3m-10)$ faulty edges, provided that each vertex of $Cay(H_n,S_n)-F$ is incident with at least two fault-free edges, where $3m-10$ is maximum number of faulty edges. Finally, we prove that $Cay(H_n,S_n)-F$ keeps strong local diagnosability property no matter how many edges are faulty, provided that each vertex of $Cay(H_n,S_n)-F$ is incident with at least four fault-free edges.

Abstract Diagnosability of a multiprocessor system is an important research topic. The system and an interconnection network have an underlying topology, which is usually presented by a graph. Under the Maeng and Malek's (MM) model, to diagnose the system, a node sends the same task to two of its neighbors, and then compares their responses. The MM$^{*}$ is a special case of the MM model and each node must test all pairs of its adjacent nodes. In 2009, Chiang and Tan (Using node diagnosability to determine $t$-diagnosability under the comparison diagnosis (cd) model. IEEE Trans. Comput., 58, 251–259) proposed a new viewpoint for fault diagnosis of the system, namely, the node diagnosability. As a new topology structure of interconnection networks, the nest graph $CK_{n}$ has many good properties. In this paper, we study the local diagnosability of $CK_{n}$ and show it has the strong local diagnosability property even if there exist $(\frac{n(n-1)}{2}-2)$ missing edges in it under the MM$^{*}$ model, and the result is optimal with respect to the number of missing edges.