The g-Good-Neighbor Diagnosability of Folded Hypercube-like Networks

Equal relation between [formula omitted]-good-neighbor diagnosability under the PMC model and [formula omitted]-good-neighbor diagnosability under the MM[formula omitted] model of a graph

Diagnosability is often considered as an important factor for measuring the self-diagnostic ability of network systems. However, classic system-level diagnosis focuses only on processor faults and ignores the objective reality of communication faults. Under real circumstances, missing edges and node failures usually occur simultaneously in multiprocessor systems (called hybrid fault circumstances). Therefore, it is important to study the diagnosability of multiprocessor systems under hybrid fault circumstances. In this paper, we propose several diagnosabilities of interconnection networks with missing edges and faulty nodes. By exploring some important relationships between diagnosability and the minimum degree of a network under hybrid fault circumstances, we present and prove the diagnosability of several classic interconnection networks, including BC (bijective connection) networks, star graphs, folded hypercubes, exchanged hypercubes, exchanged crossed cubes, k-ary n-cubes, bubble-sort star graphs and balanced hypercubes, with missing edges and broken-down nodes under the PMC (Preparata, Metze and Chien) and MM* (Maeng and Malek) models.

An interconnection network's diagnosability is an important measure of its self-diagnostic capability. In 2012, Peng et al. proposed a measure for fault diagnosis of the network, namely, the $h$-good-neighbor conditional diagnosability, which requires that every fault-free node has at least $h$ fault-free neighbors. There are two well-known diagnostic models, PMC model and MM* model. The {\it $h$-good-neighbor diagnosability} under the PMC (resp. MM*) model of a graph $G$, denoted by $t_h^{PMC}(G)$ (resp. $t_h^{MM^*}(G)$), is the maximum value of $t$ such that $G$ is $h$-good-neighbor $t$-diagnosable under the PMC (resp. MM*) model. In this paper, we study the $2$-good-neighbor diagnosability of some general $k$-regular $k$-connected graphs $G$ under the PMC model and the MM* model. The main result $t_2^{PMC}(G)=t_2^{MM^*}(G)=g(k-1)-1$ with some acceptable conditions is obtained, where $g$ is the girth of $G$. Furthermore, the following new results under the two models are obtained: $t_2^{PMC}(HS_n)=t_2^{MM^*}(HS_n)=4n-5$ for the hierarchical star network $HS_n$, $t_2^{PMC}(S_n^2)=t_2^{MM^*}(S_n^2)=6n-13$ for the split-star networks $S_n^2$ and $t_2^{PMC}(\Gamma_{n}(\Delta))=t_2^{MM^*}(\Gamma_{n}(\Delta))=6n-16$ for the Cayley graph generated by the $2$-tree $\Gamma_{n}(\Delta)$.

Diagnosability of a multiprocessor system is an important research topic. The system and interconnection network has a underlying topology, which usually presented by a graph G = ( V , E ) . In 2012, a measurement for fault tolerance of the graph was proposed by Peng et al. This measurement is called the g-good-neighbor diagnosability that restrains every fault-free node to contain at least g fault-free neighbors. Under the PMC model, to diagnose the system, two adjacent nodes in G are can perform tests on each other. Under the 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 its any pair of adjacent nodes of the system. As a famous topology structure, the ( n , k ) -arrangement graph A n , k , has many good properties. In this paper, we give the g-good-neighbor diagnosability of A n , k under the PMC model and MM* model.

Connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks. The $R^g$-vertex-connectivity of a connected graph $G$ is the minimum cardinality of a faulty set $X\subseteq V(G)$ such that $G-X$ is disconnected and every fault-free vertex has at least $g$ fault-free neighbors. The $g$-good-neighbor conditional diagnosability is defined as the maximum cardinality of a $g$-good-neighbor conditional faulty set that the system can guarantee to identify. The interconnection network considered here is the locally exchanged twisted cube $LeTQ(s,t)$. For $1\leq s\leq t$ and $0\leq g\leq s$, we first determine the $R^g$-vertex-connectivity of $LeTQ(s,t)$, then establish the $g$-good neighbor conditional diagnosability of $LeTQ(s,t)$ under the PMC model and MM$^*$ model, respectively.

The 2-good-neighbor (2-extra) diagnosability of alternating group graph networks under the PMC model and MM* model

The g-good-neighbor diagnosability of locally twisted cubes

The g-good neighbor conditional diagnosability of twisted hypercubes under the PMC and MM* model

The g-good-neighbor diagnosability of (n,k)-star graphs

Reliability evaluation of complete graph-based recursive networks

Fault tolerance is the ability such that a multiprocessor system operates properly in the event of the failure of some of its components. Latifi et al. [IEEE Trans. Comput. 43 (2) (1994) 218–222] proposed the notion of Rg-connectivity(κg) of a network modeled by graph G such that at least κg vertices in G should be deleted to disconnect the network, and the minimum degree of every connected component is at least g. This paper establishes κg(CCN(n))=(n−g+1)2g(1≤g≤n−2) for n-dimensional complete cubic network CCN(n)⁠. Fault diagnosability is another important metric for evaluating network reliability and availability. In 2012, Peng et al. [Appl. Math. Comput. 218 (21) (2012) 10406–10412] proposed a novel g-good-neighbor (conditional) diagnosability, which tacitly assumes that every fault-free vertex has at least g fault-free neighbors. In view of κg(CCN(n))⁠, we show that the g-good-neighbor diagnosability of the complete cubic network CCN(n) under the PMC model (⁠1≤g≤n−2⁠) and the MM* model (⁠1≤g≤n−2 and n≥4⁠) is (n−g+2)2g−1⁠, respectively.

The 2-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM⁎ model

Diagnosability of a multiprocessor system is an important study topic. S. L. Peng, C. K. Lin, J. J. M. Tan, and L. H. Hsu [Appl. Math. Comput., 2012, 218(21): 10406-10412] proposed a new measure for fault diagnosis of the system, which is called the g-good-neighbor conditional diagnosability that restrains every fault-free node containing at least g fault-free neighbors. As a famous topological structure of interconnection networks, the n-dimensional star graph S n has many good properties. In this paper, we establish the g-good-neighbor conditional diagnosability of S n under the PMC model and MM* model.

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 system’s diagnosability while ignoring the system’s 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.

Reliability evaluation of half hypercube networks