Reliability Evaluation of Augmented Cubes on Degree

2021 ◽  
Author(s):  
Mingzu Zhang ◽  
Xiaoli Yang ◽  
Xiaomin He ◽  
Zhuangyan Qin ◽  
Yongling Ma
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Minimum Degree ◽  
Tolerance Analysis ◽  
Reliability Evaluation ◽  
Affirmative Answer ◽  
Edge Connectivity ◽  
Minimum Number ◽  
Edge Fault Tolerance ◽  
Augmented Cube

The [Formula: see text]-dimensional augmented cube [Formula: see text], proposed by Choudum and Sunitha in 2002, is one of the most famous interconnection networks of the distributed parallel system. Reliability evaluation of underlying topological structures is vital for fault tolerance analysis of this system. As one of the most extensively studied parameters, the [Formula: see text]-conditional edge-connectivity of a connected graph [Formula: see text], [Formula: see text], is defined as the minimum number of the cardinality of the edge-cut of [Formula: see text], if exists, whose removal disconnects this graph and keeps each component of [Formula: see text] having minimum degree at least [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text] be three integers, where [Formula: see text], if [Formula: see text] and [Formula: see text], if [Formula: see text]. In this paper, we determine the exact value of the [Formula: see text]-conditional edge-connectivity of [Formula: see text], [Formula: see text] for each positive integer [Formula: see text] and [Formula: see text], and give an affirmative answer to Shinde and Borse’s corresponding conjecture on this topic in [On edge-fault tolerance in augmented cubes, J. Interconnection Netw. 20(4) (2020), DOI:10.1142/S0219265920500139].

Component Edge Connectivity of Hypercubes

2018 ◽  
Vol 29 (06) ◽  
pp. 995-1001 ◽  
Author(s):  
Shuli Zhao ◽  
Weihua Yang ◽  
Shurong Zhang ◽  
Liqiong Xu
Keyword(s):  
Fault Tolerance ◽  
Complete Graph ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Optimal Solutions ◽  
Edge Connectivity ◽  
Important Measure ◽  
Minimum Number ◽  
Text Component

Fault tolerance is an important issue in interconnection networks, and the traditional edge connectivity is an important measure to evaluate the robustness of an interconnection network. The component edge connectivity is a generalization of the traditional edge connectivity. The [Formula: see text]-component edge connectivity [Formula: see text] of a non-complete graph [Formula: see text] is the minimum number of edges whose deletion results in a graph with at least [Formula: see text] components. Let [Formula: see text] be an integer and [Formula: see text] be the decomposition of [Formula: see text] such that [Formula: see text] and [Formula: see text] for [Formula: see text]. In this note, we determine the [Formula: see text]-component edge connectivity of the hypercube [Formula: see text], [Formula: see text] for [Formula: see text]. Moreover, we classify the corresponding optimal solutions.

scholarly journals The Edge Connectivity of Expanded k-Ary n-Cubes

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Shiying Wang ◽  
Mujiangshan Wang
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Complex Problem ◽  
Multiprocessor System ◽  
Multiprocessor Systems ◽  
Complex Problem Solving ◽  
Edge Connectivity ◽  
Communication Links ◽  
Minimum Number

Mass data processing and complex problem solving have higher and higher demands for performance of multiprocessor systems. Many multiprocessor systems have interconnection networks as underlying topologies. The interconnection network determines the performance of a multiprocessor system. The network is usually represented by a graph where nodes (vertices) represent processors and links (edges) represent communication links between processors. For the network G, two vertices u and v of G are said to be connected if there is a (u,v)-path in G. If G has exactly one component, then G is connected; otherwise G is disconnected. In the system where the processors and their communication links to each other are likely to fail, it is important to consider the fault tolerance of the network. For a connected network G=(V,E), its inverse problem is that G-F is disconnected, where F⊆V or F⊆E. The connectivity or edge connectivity is the minimum number of F. Connectivity plays an important role in measuring the fault tolerance of the network. As a topology structure of interconnection networks, the expanded k-ary n-cube XQnk has many good properties. In this paper, we prove that (1) XQnk is super edge-connected (n≥3); (2) the restricted edge connectivity of XQnk is 8n-2 (n≥3); (3) XQnk is super restricted edge-connected (n≥3).

On Edge-Fault Tolerance in Augmented Cubes

2020 ◽  
Vol 20 (04) ◽  
pp. 2050013
Author(s):  
AMRUTA SHINDE ◽  
Y. M. BORSE
Keyword(s):  
Fault Tolerance ◽  
Upper Bound ◽  
Edge Connectivity ◽  
Edge Fault Tolerance ◽  
Augmented Cube

The augmented cube AQn is one of the important variations of the hypercube Qn. In this paper, we prove that the conditional h-edge connectivity of AQn with n ≥ 3 is 8n − 16 for h = 3 and 2n for h = 2n − 3. We also obtain an upper bound on the conditional h-edge connectivity for odd integer h satisfying [Formula: see text].

Analysis on the Component Connectivity of Enhanced Hypercubes

2020 ◽  
Author(s):  
Liqiong Xu ◽  
Litao Guo
Keyword(s):  
Connected Graph ◽  
Interconnection Networks ◽  
Unsolved Problem ◽  
Reliability Evaluation ◽  
Minimum Number ◽  
Appl Math

Abstract Reliability evaluation of interconnection networks is of significant importance to the design and maintenance of interconnection networks. The component connectivity is an important parameter for the reliability evaluation of interconnection networks and is a generalization of the traditional connectivity. The $g$-component connectivity $c\kappa _g (G)$ of a non-complete connected graph $G$ is the minimum number of vertices whose deletion results in a graph with at least $g$ components. Determining the $g$-component connectivity is still an unsolved problem in many interconnection networks. Let $Q_{n,k}$ ($1\leq k\leq n-1$) denote the $(n, k)$-enhanced hypercube. In this paper, let $n\geq 7$ and $1\leq k \leq n-5$, we determine $c\kappa _{g}(Q_{n,k}) = g(n + 1) - \frac{1}{2}g(g + 1) + 1$ for $2 \leq g \leq n$. The previous result in Zhao and Yang (2019, Conditional connectivity of folded hypercubes. Discret. Appl. Math., 257, 388–392) is extended.

Structure Fault Tolerance of Recursive Interconnection Networks

2020 ◽  
Author(s):  
Eminjan Sabir ◽  
Jixiang Meng
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Theoretical Problem ◽  
Connected Subgraph ◽  
Edge Connectivity ◽  
Minimum Cardinality ◽  
Edge Disjoint ◽  
Balanced Hypercube

Abstract Motivated by effects caused by structure link faults in networks, we study the following graph theoretical problem. Let $T$ be a connected subgraph of a graph $G$ except for $K_{1}$. The $T$-structure edge-connectivity $\lambda (G;T)$ (resp. $T$-substructure edge-connectivity $\lambda ^s(G;T)$) of $G$ is the minimum cardinality of a set of edge-disjoint subgraphs $\mathcal{F}=\{T_{1},T_{2},\ldots ,T_{m}\}$ (resp. $\mathcal{F}=\{T_{1}^{^{\prime}},T_{2}^{^{\prime}},\ldots ,T_{m}^{^{\prime}}\}$) such that $T_{i}$ is isomorphic to $T$ (resp. $T_{i}^{^{\prime}}$ is a connected subgraph of $T$) for every $1 \le i \le m$, and $E(\mathcal{F})$’s removal leaves the remaining graph disconnected. In this paper, we determine both $\lambda (G;T)$ and $\lambda ^{s}(G;T)$ for $(1)$ the hypercube $Q_{n}$ and $T\in \{K_{1,1},K_{1,2},K_{1,3},P_{4},Q_{1},Q_{2},Q_{3}\}$; $(2)$ the $k$-ary $n$-cube $Q^{k}_{n}$$(k\ge 3)$ and $T\in \{K_{1,1},K_{1,2},K_{1,3},Q^{3}_{1},Q^{4}_{1}\}$; $(3)$ the balanced hypercube $BH_{n}$ and $T\in \{K_{1,1},K_{1,2},BH_{1}\}$. We also extend some known results.

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