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.

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The ${\schmi g}$ -Good-Neighbor Conditional Diagnosability of ${\schmi k}$ -Ary ${\schmi n}$ -Cubes under the PMC Modeland MM* Model

Given a graph ${\mbi{G}}$ and a non-negative integer ${{g}}$ , the ${{g}}$ -extra connectivity (resp. ${{g}}$ -extra edge connectivity) of ${\mbi{G}}$ is the minimum cardinality of a set of vertices (resp. edges) in ${\mbi{G}}$ , if it exists, whose deletion disconnects ${\mbi{G}}$ and leaves each remaining component with more than ${{g}}$ vertices. This study shows that the 3-extra connectivity (resp. 3-extra edge connectivity) of an ${\mbi{n}}$ -dimensional folded hypercube is ${4}{{n}} - {5}$ for ${{n}} \geq {6}$ (resp. ${4}{{n}} - {4}$ for ${{n}} \geq {5}$ ). This study also provides an upper bound for the ${{g}}$ -extra connectivity on folded hypercubes for ${{g}} \geq {6}$ .

On 3-Extra Connectivity and 3-Extra Edge Connectivity of Folded Hypercubes

Processor fault diagnosis has played an important role in measuring the reliability of a multiprocessor system, and the diagnosability of many well-known multiprocessor systems has been widely investigated. The conditional diagnosability is a novel measure of diagnosability by adding an additional condition that any faulty set cannot contain all the neighbors of any node in a system. In this paper, we evaluate the conditional diagnosability for augmented cubes under the PMC model. We show that the conditional diagnosability of an n-dimensional augmented cube is 8n - 27 for n≥5.

Conditional Diagnosability of Augmented Cubes under the PMC Model

The conditional diagnosis is a very important measure of the reliability and the fault-tolerance of networks. The ‘condition’ means that no faulty set contains all neighbours of any node. Under this assumption, for any system G, every component of has more than 1 node, where F is the faulty set of G. The g-extra conditional diagnosability is defined under the assumption that every component of has more than nodes. ‘A system with at most t faulty nodes is defined as sequentially t-diagnosable if at least one faulty node can be repaired, so that the testing can be continued using the repaired node to eventually diagnose all faulty nodes’ [E.P. Duarte Jr., R.P. Ziwich, and L.C.P. Albini, A survey of comparison-based system-level diagnosis, ACM Comput. Surv. 433 2011, article 22]. To increase the degree of the sequential t-diagnosability of a system, sequential -diagnosis strategy is proposed in this paper. It is allowed that there are at most k misdiagnosed nodes. In this paper, we determine the g-extra conditional diagnosability of hypercubes and propose sequential -diagnosis algorithms for hypercubes with low time complexities under the Preparata, Metze, and Chien PMC model and the MM* model which is a special case of the Maeng and Malek MM model.

The g-extra conditional diagnosability and sequential t/k-diagnosability of hypercubes

To find a cycle (resp. path) of a given length in a graph is the cycle (resp. path) embedding problem. To find cycles of all lengths from its girth to its order in a graph is the pancyclic problem. A stronger concept than the pancylicity is the panconnectivity. A graph of order n is said to be panconnected if for any pair of different vertices x and y with distance d there exist xy-paths of every length from d to n. The pancyclicity or the panconnectivity is an important property to determine if the topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The pancyclicity and the panconnectivity of interconnection networks have attracted much research interest in recent years. A large amount of related work appeared in the literature, with some repetitions. The purpose of this paper is to give a survey of the results related to these topics for the hypercube and some hypercube-like networks.

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Survey on path and cycle embedding in some networks

https://ir.nctu.edu.tw:443/bitstream/11536/7665/1/000264234500006.pdf

Conditional diagnosability of hypercubes under the comparison diagnosis model

https://ir.nctu.edu.tw:443/bitstream/11536/9452/1/000255072500004.pdf

Strong Menger connectivity with conditional faults on the class of hypercube-like networks

https://ir.nctu.edu.tw/bitstream/11536/79134/1/360401.pdf

Edge-bipancyclicity of conditional faulty hypercubes

The local connectivity of two vertices is defined as the maximum number of internally vertex-disjoint paths between them. In this paper, we define two vertices as maximally local-connected, if the maximum number of internally vertex-disjoint paths between them equals the minimum degree of these two vertices. Moreover, we show that an (n-1)-regular Cayley graph generated by transposition tree is maximally local-connected, even if there are at most (n-3) faulty vertices in it, and prove that it is also (n-1)-fault-tolerant one-to-many maximally local-connected.

Fault-tolerant maximal local-connectivity on cayley graphs generated by transposition trees

An interconnection network is usually modeled as a graph, in which vertices and edges correspond to processor and communication links, respectively. The local connectivity of two vertices is defined as the maximum number of internally vertex-disjoint paths between them. In this paper, we define two vertices to be maximally local-connected, if the maximum number of internally vertex-disjoint paths between them equals the minimum degree of these two vertices. Moreover, we introduce the one-to-many version of connectivity. We show that an n-dimensional Bubble-sort Graph is maximally local-connected, even if there are at most n − 3 faulty vertices in it, and prove that it is also (n − 1)-fault-tolerant one-to-many maximally local-connected.

Lun-Min Shih

Papers

Conditional diagnosability of hypercubes under the comparison diagnosis model

Strong Menger connectivity with conditional faults on the class of hypercube-like networks

Edge-bipancyclicity of conditional faulty hypercubes

Fault-tolerant maximal local-connectivity on cayley graphs generated by transposition trees

Fault-Tolerant Maximal Local-Connectivity on the Bubble-Sort Graphs