Endogenous security has the ability to discover, eliminate, and solve internal security problems and hidden dangers within the network, and is a superior technology to ensure future network security. The$t/k$-diagnosis strategy, as a strong and adaptive self-diagnosis strategy, is an important part for ensuring endogenous security. Moreover, the folded hypercube ($FQ_{n}$) as a data transmission network (e.g., optical network) topology offers new potential for the construction of large-scale, high data throughput, and low-latency systems, such as computing network, human-cyber-physical systems, and smart grid. However, there are few studies on endogenous security based on$FQ_{n}$networks. Therefore, this article designs an adaptive system-level fault self-diagnosis strategy, namely Fast t/k-Diagnosis Under Maeng-Malek Model (Ftk-DIAG-MM*) to diagnosis the faulty vertices in$FQ_{n}$network under the Maeng-Malek model (MM* mod). Then, we provide a proof of the algorithm correctness theoretically by the fault tolerance of$FQ_{n}$network. It is derived by theoretical derivation that the$t/k$-diagnosability inherent to the$FQ_{n}$network is$(n+1)\break(k+1)-k(k+3)/2$. The simulation experiments demonstrate that the designed Ftk-DIAG-MM* strategy can correctly diagnose all vertices within the range allowed by the diagnosability, and still has a great performance when it exceeds the range allowed by the diagnosability. It greatly enhances the fault diagnosis capability of$FQ_{n}$network in the circumstance of misdiagnosing a few vertices, which provides an important theoretical basis for the reliability and endogenous security of$FQ_{n}$networks.

In order to evaluate the reliability of a multiprocessor system, the fault diagnosability was introduced and utilized as a significant indicator. In the study of fault diagnosability, researchers usually concentrate on the diagnosability of the global system but ignore its local information. However, the local information also plays a crucial role in the reliability of a multiprocessor system. Thus, an innovative concept of fault diagnosability, called$y$-fault-free-block local fault diagnosability, is put forward to study the fault diagnosability of a multiprocessor system at local vertex, where the$y$-fault-free-block condition requires more than$y$connected vertices. In this article, we characterize several important properties about the$y$-fault-free-block local fault diagnosability of a multiprocessor interconnection network under the MM* model and propose its$y$-fault-free-block local fault diagnosability at local vertex. Furthermore, we apply our results to some well-known networks, and we obtain their$y$-fault-free-block local fault diagnosabilities at local vertex directly under the MM* model, including bijective connection graph, star graph, and$(n,k)$-star graph. Finally, we compare the$y$-fault-free-block local fault diagnosability of a graph at local vertex with other types of diagnosability, including the diagnosability, conditional diagnosability, good-neighbor diagnosability, and pessimistic diagnosability. It can be seen that the$y$-fault-free-block local fault diagnosability at vertex is larger than all the other types of diagnosability.

Fault diagnosability is utilized as a significant measure that reflects the reliability of a multiprocessor system. However, people frequently pay close attention to the entire systems diagnosability while ignoring the systems important local information. The m-fault-free-neighbor local fault diagnosability (for short, m-FFNLFD) is a novel indicator, which describes the diagnosability of a system at a local node with m fault-free neighbors. In this paper, we propose the m-FFNLFD of general networks at local node under the Preparata Metze Chien model. Moreover, we also characterize some important properties of m-FFNLFD of a multiprocessor system under the comparison model. Furthermore, we apply our proposed conclusions to directly obtain the m-FFNLFD of 11 well-known networks under PMC-M and MM*-M, including hypercubes, locally twisted cubes, k-ary n-cubes, crossed cubes, twisted hypercubes, exchanged hypercubes, star graphs, (n, k)-star graphs, (n, k)-arrangement graphs, data center network DCells and BCDCs. Finally, we compare the m-FFNLFD with both diagnosability and conditional diagnosability, and it is shown that the m-FFNLFD is greater than all the other fault diagnosabilities.

For interconnection network losing processors, usually, when the surviving network has a large connected component, it can be used as a functional subsystem without leading to severe performance degradation. Consequently, it is crucial to characterize the interprocessor communication ability and efficiency of the surviving structure. In this article, we prove that when a subset$D$of at most$6n-17$processors is deleted from an$n$-dimensional alternating group graph$\text{AG}_n$, there exists a largest component with cardinality greater or equal to$|V(\text{AG}_n)|-|D|-3$for$n\geq 6$in the remaining network, and the union of small components is, first, an empty graph; or, second, a 3-cycle, or an edge, or a 2-path, or a singleton; or, third, an edge and a singleton, or two singletons. Then, we prove that when a subset$D$of at most$8n-25$processors is deleted from$\text{AG}_n$, there exists a largest component with cardinality greater or equal to$|V(\text{AG}_n)|-|D|-5$for$n\geq 6$in the remaining network, and the union of small components is, first, an empty graph; or, second, a 5-cycle, or a 4-path, or a 4-claw, or a 4-cycle, or a 3-path, or a 3-claw, or a 3-cycle, or a 2-path, or an edge, or a singleton; or, third, a 4-cycle and a singleton, or a 3-path and a singleton, or a 3-claw and a singleton, or a 2-path and a singleton, two edges, an edge and a singleton, or two singletons; or, fourth, two edges and a singleton, or a 2-path and two singletons, or an edge and two singletons, or three singletons.