Hybrid Fault diagnosis capability analysis of Hypercubes under the PMC model and MM* model

System level diagnosis is an important approach for the fault diagnosis of multiprocessor systems. In system level diagnosis, diagnosability is an important measure of the diagnosis capability of interconnection networks. But as a measure, diagnosability can not reflect the diagnosis capability of multiprocessor systems to link faults which may occur in real circumstances. In this paper, we propose the definition of $h$-edge tolerable diagnosability to better measure the diagnosis capability of interconnection networks under hybrid fault circumstances. The $h$-edge tolerable diagnosability of a multiprocessor system $G$ is the maximum number of faulty nodes that the system can guarantee to locate when the number of faulty edges does not exceed $h$,denoted by $t_h^{e}(G)$. The PMC model and MM model are the two most widely studied diagnosis models for the system level diagnosis of multiprocessor systems. The hypercubes are the most well-known interconnection networks. In this paper, the $h$-edge tolerable diagnosability of $n$-dimensional hypercube under the PMC model and MM$^{*}$ is determined as follows: $t_h^{e}(Q_n)= n-h$, where $1\leq h<n$, $n\geq3$.Comment: 5 pages, 1 figur

Conditional Fault Diagnosis of Bubble Sort Graphs under the PMC Model

As the size of a multiprocessor system increases, processor failure is inevitable, and fault identification in such a system is crucial for reliable computing. The fault diagnosis is the process of identifying faulty processors in a multiprocessor system through testing. For the practical fault diagnosis systems, the probability that all neighboring processors of a processor are faulty simultaneously is very small, and the conditional diagnosability, which is a new metric for evaluating fault tolerance of such systems, assumes that every faulty set does not contain all neighbors of any processor in the systems. This paper shows that the conditional diagnosability of bubble sort graphs $B_n$ under the PMC model is $4n-11$ for $n \geq 4$, which is about four times its ordinary diagnosability under the PMC model

The gg-good neighbor conditional diagnosability of locally exchanged twisted cubes

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.Comment: 19 pages, 4 figure

The gg-good neighbour diagnosability of hierarchical cubic networks

Let $G=(V, E)$ be a connected graph, a subset $S\subseteq V(G)$ is called an $R^{g}$-vertex-cut of $G$ if $G-F$ is disconnected and any vertex in $G-F$ has at least $g$ neighbours in $G-F$. The $R^{g}$-vertex-connectivity is the size of the minimum $R^{g}$-vertex-cut and denoted by $\kappa^{g}(G)$. Many large-scale multiprocessor or multi-computer systems take interconnection networks as underlying topologies. Fault diagnosis is especially important to identify fault tolerability of such systems. The $g$-good-neighbor diagnosability such that every fault-free node has at least $g$ fault-free neighbors is a novel measure of diagnosability. In this paper, we show that the $g$-good-neighbor diagnosability of the hierarchical cubic networks $HCN_{n}$ under the PMC model for $1\leq g\leq n-1$ and the $MM^{*}$ model for $1\leq g\leq n-1$ is $2^{g}(n+2-g)-1$, respectively

Relationship between Conditional Diagnosability and 2-extra Connectivity of Symmetric Graphs

The conditional diagnosability and the 2-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault-tolerance in a multiprocessor system. The conditional diagnosability $t_c(G)$ of $G$ is the maximum number $t$ for which $G$ is conditionally $t$-diagnosable under the comparison model, while the 2-extra connectivity $\kappa_2(G)$ of a graph $G$ is the minimum number $k$ for which there is a vertex-cut $F$ with $|F|=k$ such that every component of $G-F$ has at least $3$ vertices. A quite natural problem is what is the relationship between the maximum and the minimum problem? This paper partially answer this problem by proving $t_c(G)=\kappa_2(G)$ for a regular graph $G$ with some acceptable conditions. As applications, the conditional diagnosability and the 2-extra connectivity are determined for some well-known classes of vertex-transitive graphs, including, star graphs, $(n,k)$-star graphs, alternating group networks, $(n,k)$-arrangement graphs, alternating group graphs, Cayley graphs obtained from transposition generating trees, bubble-sort graphs, $k$-ary $n$-cube networks and dual-cubes. Furthermore, many known results about these networks are obtained directly

The tightly super 3-extra connectivity and 3-extra diagnosability of crossed cubes

Many multiprocessor systems have interconnection networks as underlying topologies and an interconnection network is usually represented by a graph where nodes represent processors and links represent communication links between processors. In 2016, Zhang et al. proposed the $g$-extra diagnosability of $G$, which restrains that every component of $G-S$ has at least $(g +1)$ vertices. As an important variant of the hypercube, the $n$-dimensional crossed cube $CQ_{n}$ has many good properties. In this paper, we prove that $CQ_{n}$ is tightly $(4n-9)$ super 3-extra connected for $n\geq 7$ and the 3-extra diagnosability of $CQ_{n}$ is $4n-6$ under the PMC model $(n\geq5)$ and MM$^*$ model $(n\geq7)$

On the gg-good-neighbor connectivity of graphs

Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, F\`{a}brega and Fiol proposed the $g$-good-neighbor connectivity of $G$. In this paper, we show that $1\leq \kappa^g(G)\leq n-2g-2$ for $0\leq g\leq \left\{\Delta(G),\left\lfloor \frac{n-3}{2}\right\rfloor\right\}$, and graphs with $\kappa^g(G)=1,2$ and trees with $\kappa^g(T_n)=n-t$ for $4\leq t\leq \frac{n+2}{2}$ are characterized, respectively. In the end, we get the three extremal results for the $g$-good-neighbor connectivity.Comment: 14 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1904.06527; text overlap with arXiv:1609.08885, arXiv:1612.05381 by other author

Cycles in enhanced hypercubes

The enhanced hypercube $Q_{n,k}$ is a variant of the hypercube $Q_n$. We investigate all the lengths of cycles that an edge of the enhanced hypercube lies on. It is proved that every edge of $Q_{n,k}$ lies on a cycle of every even length from $4$ to $2^n$; if $k$ is even, every edge of $Q_{n,k}$ also lies on a cycle of every odd length from $k+3$ to $2^n-1$, and some special edges lie on a shortest odd cycle of length $k+1$.Comment: 9 pages, 2 figure

The Component Connectivity of Alternating Group Graphs and Split-Stars

For an integer $\ell\geqslant 2$, the $\ell$-component connectivity of a graph $G$, denoted by $\kappa_{\ell}(G)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a graph with fewer than $\ell$ vertices. This is a natural generalization of the classical connectivity of graphs defined in term of the minimum vertex-cut and is a good measure of robustness for the graph corresponding to a network. So far, the exact values of $\ell$-connectivity are known only for a few classes of networks and small $\ell$'s. It has been pointed out in~[Component connectivity of the hypercubes, Int. J. Comput. Math. 89 (2012) 137--145] that determining $\ell$-connectivity is still unsolved for most interconnection networks, such as alternating group graphs and star graphs. In this paper, by exploring the combinatorial properties and fault-tolerance of the alternating group graphs $AG_n$ and a variation of the star graphs called split-stars $S_n^2$, we study their $\ell$-component connectivities. We obtain the following results: (i) $\kappa_3(AG_n)=4n-10$ and $\kappa_4(AG_n)=6n-16$ for $n\geqslant 4$, and $\kappa_5(AG_n)=8n-24$ for $n\geqslant 5$; (ii) $\kappa_3(S_n^2)=4n-8$, $\kappa_4(S_n^2)=6n-14$, and $\kappa_5(S_n^2)=8n-20$ for $n\geqslant 4$

The vulnerability of the diameter of enhanced hypercubes

For an interconnection network $G$, the {\it $\omega$-wide diameter} $d_\omega(G)$ is the least $\ell$ such that any two vertices are joined by $\omega$ internally-disjoint paths of length at most $\ell$, and the {\it $(\omega-1)$-fault diameter} $D_{\omega}(G)$ is the maximum diameter of a subgraph obtained by deleting fewer than $\omega$ vertices of $G$. The enhanced hypercube $Q_{n,k}$ is a variant of the well-known hypercube. Yang, Chang, Pai, and Chan gave an upper bound for $d_{n+1}(Q_{n,k})$ and $D_{n+1}(Q_{n,k})$ and posed the problem of finding the wide diameters and fault diameters of $Q_{n,k}$. By constructing internally disjoint paths between any two vertices in the enhanced hypercube, for $n\ge3$ and $2\le k\le n$ we prove $$D_\omega(Q_{n,k})=d_\omega(Q_{n,k})=\begin{cases} d(Q_{n,k}) & \textrm{for $1 \leq \omega < n-\lfloor\frac{k}{2}\rfloor$;}\\ d(Q_{n,k})+1 & \textrm{for $n-\lfloor\frac{k}{2}\rfloor \leq \omega \leq n+1$.} \end{cases}$$ where $d(Q_{n,k})$ is the diameter of $Q_{n,k}$. These results mean that interconnection networks modelled by enhanced hypercubes are extremely robust.Comment: 9 pages, 1 figur