Abstract Motivated by $g$-extra connectivity, the $g$-extra diagnosability is proposed as a better and more realistic measurement for fault diagnosis of interconnection networks, which is defined as the maximum number of faulty vertices that can be identified when each remaining component has no fewer $g+1$ vertices. Under the MM* model, a variety of interconnection networks’ $g$-extra diagnosability have been investigated, such as hypercube, folded hypercube, $(n,k)$-star network, alternating group graph, etc. These results mostly share similar derivation processes to derive the $g$-extra diagnosability of involved networks by using the $g$-extra connectivity. Therefore, a general approach to derive the $g$-extra diagnosability of a network from its $g$-extra connectivity was investigated in (Wang, S. Y. and Wang, M. (2019) The $g$-good-neighbor and $g$-extra diagnosability of networks. Theor. Comput. Sci., 773, 107–114) and (Huang, Y. Z., Lin, L. M. and Xu, L. (2020) A new proof for exact relationship between extra connectivity and extra diagnosability of regular connected graphs under MM* model. Theor. Comput. Sci., 828–829, 70–80). However, there are some shortcomings in both references. By summarizing the existing shared practices, we propose a new relationship between the $g$-extra connectivity and the $g$-extra diagnosability of networks under the MM* model. As applications, we derive the $g$-extra diagnosability of bijective connection networks and $(n,k)$-star graphs.

The diagnosability of a system is defined as the maximum number of faulty processors that the system can guarantee to identify, which plays an important role in measuring of the reliability of multiprocessor systems. In the work of Peng et al. in 2012, they proposed a new measure for fault diagnosis of systems, namely,$g$-good-neighbor conditional diagnosability. It is defined as the diagnosability of a multiprocessor system under the assumption that every fault-free node contains at least$g$fault-free neighbors, which can measure the reliability of interconnection networks in heterogeneous environments more accurately than traditional diagnosability. The$k$-ary$n$-cube is a family of popular networks. In this study, we first investigate and determine the$R_g$-connectivity of$k$-ary$n$-cube for$0\le g\le n.$Based on this, we determine the$g$-good-neighbor conditional diagnosability of$k$-ary$n$-cube under the PMC model and MM* model for$k\ge 4, n\ge 3$and$0\le g\le n.$Our study shows the$g$-good-neighbor conditional diagnosability of$k$-ary$n$-cube is several times larger than the classical diagnosability of$k$-ary$n$-cube.