Relation of Extra Edge Connectivity and Component Edge Connectivity for Regular Networks

Reliability of interconnection networks is important to design multiprocessor systems. The extra edge connectivity and component edge connectivity are two parameters for the reliability evaluation. The [Formula: see text]-extra edge connectivity [Formula: see text] is the cardinality of the minimum extra edge cut [Formula: see text] such that [Formula: see text] is not connected and each component of [Formula: see text] has at least [Formula: see text] vertices. The [Formula: see text]-component edge connectivity [Formula: see text] of a graph [Formula: see text] is the minimum edge number of a set [Formula: see text] such that [Formula: see text] is not connected and [Formula: see text] has at least [Formula: see text] components. In this paper, we find the relation of extra edge connectivity and component edge connectivity for regular networks. As an application, we determine the component edge connectivity of BC networks, [Formula: see text]-ary [Formula: see text]-cubes, enhanced hypercubes.

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Component Edge Connectivity of Hypercubes

International Journal of Foundations of Computer Science ◽

10.1142/s012905411850017x ◽

2018 ◽

Vol 29 (06) ◽

pp. 995-1001 ◽

Cited By ~ 9

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.

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The Edge Connectivity of Expanded k-Ary n-Cubes

Discrete Dynamics in Nature and Society ◽

10.1155/2018/7867342 ◽

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).

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Reliability Evaluation of Augmented Cubes on Degree

Journal of Interconnection Networks ◽

10.1142/s0219265921420196 ◽

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].

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Super Ck and Sub-Ck Connectivity of k-Ary n-Cube Networks

International Journal of Foundations of Computer Science ◽

10.1142/s0129054121500088 ◽

2021 ◽

pp. 1-12

Author(s):

Yuxing Yang

Keyword(s):

Interconnection Networks ◽

Undirected Graph ◽

Text Structure ◽

Multiprocessor Systems ◽

Cube Formula

Let [Formula: see text] be an undirected graph. An H-structure-cut (resp. H-substructure-cut) of [Formula: see text] is a set of subgraphs of [Formula: see text], if any, whose deletion disconnects [Formula: see text], where the subgraphs deleted are isomorphic to a certain graph [Formula: see text] (resp. where for any [Formula: see text] of the subgraphs deleted, there is a subgraph [Formula: see text] of [Formula: see text], isomorphic to [Formula: see text], such that [Formula: see text] is a subgraph of [Formula: see text]). [Formula: see text] is super [Formula: see text]-connected (resp. super sub-[Formula: see text]-connected) if the deletion of an arbitrary minimum [Formula: see text]-structure-cut (resp. minimum [Formula: see text]-substructure-cut) isolates a component isomorphic to a certain graph [Formula: see text]. The [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is one of the most attractive interconnection networks for multiprocessor systems. In this paper, we prove that [Formula: see text] with [Formula: see text] is super sub-[Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd, and super [Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd.

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Connectivity and Nature Diagnosability of Leaf-Sort Graphs

Journal of Interconnection Networks ◽

10.1142/s0219265920500115 ◽

2020 ◽

Vol 20 (03) ◽

pp. 2050011

Author(s):

JUTAO ZHAO ◽

SHIYING WANG

Keyword(s):

Interconnection Networks ◽

Interconnection Network ◽

Research Topic ◽

Multiprocessor System ◽

Important Research ◽

Edge Connectivity ◽

Topology Structure ◽

Pmc Model ◽

Important Research Topic

The connectivity and diagnosability of a multiprocessor system or an interconnection network is an important research topic. The system and interconnection network has a underlying topology, which usually presented by a graph. As a famous topology structure of interconnection networks, the n-dimensional leaf-sort graph CFn has many good properties. In this paper, we prove that (a) the restricted edge connectivity of CFn (n ≥ 3) is 3n − 5 for odd n and 3n − 6 for even n; (b) CFn (n ≥ 5) is super restricted edge-connected; (c) the nature diagnosability of CFn (n ≥ 4) under the PMC model is 3n − 4 for odd n and 3n − 5 for even n; (d) the nature diagnosability of CFn (n ≥ 5) under the MM* model is 3n − 4 for odd n and 3n − 5 for even n.

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Analysis on the Component Connectivity of Enhanced Hypercubes

The Computer Journal ◽

10.1093/comjnl/bxaa122 ◽

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.

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K-PROCESSOR RELIABILITY OF LARGE-SCALE RING-BASED HIERARCHICAL INTERCONNECTIONS

International Journal of Reliability Quality and Safety Engineering ◽

10.1142/s0218539302000664 ◽

2002 ◽

Vol 09 (01) ◽

pp. 61-77

Author(s):

M. AL-ROUSAN ◽

O. AL-JARRAH ◽

M. MOWAFI

Keyword(s):

Interconnection Networks ◽

Large Scale ◽

Network Reliability ◽

Multiprocessor Systems ◽

Hierarchical Systems ◽

Hierarchical Networks ◽

Coherent Interface ◽

Large Acceptance ◽

Ring Architecture ◽

Scalable Coherent Interface

Recently, connecting thousands of processors via interconnection networks based on multiple (hierarchical) rings has an increased interest. This is due to the large acceptance and success of the Scalable Coherent Interface (SCI) technology. The inherently weak behavior of ring architecture has led interconnection designers to consider various choices to improve the overall network reliability. An interesting choice is to use braided rings instead of the single (basic) rings in the hierarchy. In this paper, we present new formulas for computing K-processor reliability of SCI ring-based hierarchical networks in the context of large-scale multiprocessor systems. The derived formulas are general and applicable to any given systems size consisting of an arbitrary number of levels. The reliability of hierarchical systems based on the basic and braided rings is evaluated and analyzed using the derived formulas. The results show that hierarchical systems based on braided rings significantly improve the reliability of hierarchies constructed of basic rings. The results are general and not limited to systems of SCI rings; the analysis is valid for any type of rings architecture such as token and slotted rings.

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