The Tightly Super 2-extra Connectivity and 2-extra Diagnosability of Locally Twisted Cubes

Connectivity plays an important role in measuring the fault tolerance of an interconnection network [Formula: see text]. A faulty set [Formula: see text] is called a g-extra faulty set if every component of G − F has more than g nodes. A g-extra cut of G is a g-extra faulty set F such that G − F is disconnected. The minimum cardinality of g-extra cuts is said to be the g-extra connectivity of G. G is super g-extra connected if every minimum g-extra cut F of G isolates one connected subgraph of order g + 1. If, in addition, G − F has two components, one of which is the connected subgraph of order g + 1, then G is tightly [Formula: see text] super g-extra connected. Diagnosability is an important metric for measuring the reliability of G. A new measure for fault diagnosis of G restrains that every fault-free component has at least (g + 1) fault-free nodes, which is called the g-extra diagnosability of G. The locally twisted cube LTQn is applied widely. In this paper, it is proved that LTQn is tightly (3n − 5) super 2-extra connected for [Formula: see text], and the 2-extra diagnosability of LTQn is 3n − 3 under the PMC model ([Formula: see text]) and MM* model ([Formula: see text]).

Download Full-text

The r-Extra Diagnosability of Hyper Petersen Graphs

International Journal of Foundations of Computer Science ◽

10.1142/s0129054121500234 ◽

2021 ◽

pp. 1-12

Author(s):

Shiying Wang

Keyword(s):

Fault Tolerance ◽

Fault Diagnosis ◽

Interconnection Networks ◽

Interconnection Network ◽

Petersen Graph ◽

Multiprocessor System ◽

Topology Structure ◽

Pmc Model ◽

Petersen Graphs ◽

Text Model

The diagnosability of a multiprocessor system or an interconnection network plays an important role in measuring the fault tolerance of the network. In 2016, Zhang et al. proposed a new measure for fault diagnosis of the system, namely, the [Formula: see text]-extra diagnosability, which restrains that every fault-free component has at least [Formula: see text] fault-free nodes. As a famous topology structure of interconnection networks, the hyper Petersen graph [Formula: see text] has many good properties. It is difficult to prove the [Formula: see text]-extra diagnosability of an interconnection network. In this paper, we show that the [Formula: see text]-extra diagnosability of [Formula: see text] is [Formula: see text] for [Formula: see text] and [Formula: see text] in the PMC model and for [Formula: see text] and [Formula: see text] in the MM[Formula: see text] model.

Download Full-text

Embedding of Binomial Trees in Locally Twisted Cubes with Link Faults

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.1049-1050.1736 ◽

2014 ◽

Vol 1049-1050 ◽

pp. 1736-1740

Author(s):

Lan Tao You

Keyword(s):

Parallel Computing ◽

Interconnection Network ◽

Binomial Tree ◽

Binomial Trees ◽

Locally Twisted Cubes ◽

Twisted Cube ◽

Twisted Cubes

As a newly introduced interconnection network for parallel computing, the locally twisted cube possesses many desirable properties. In this paper, binomial tree embeddings in locally twisted cubes are studied. We present two major results in this paper: (1) For any integern≥ 2, ann-dimensional binomial treeBncan be embedded inLTQnwith dilation 1 by randomly choosing any vertex inLTQnas the root. (2) For any integern≥ 2, ann-dimensional binomial treeBncan be embedded inLTQnwith up ton− 1 faulty links inlog(n− 1) steps where dilation = 1. The results are optimal in the sense that the dilations of all embeddings are 1.

Download Full-text

Embedding Complete Binary Trees into Locally Twisted Cubes

Advanced Engineering Forum ◽

10.4028/www.scientific.net/aef.6-7.70 ◽

2012 ◽

Vol 6-7 ◽

pp. 70-75 ◽

Cited By ~ 4

Author(s):

Yue Juan Han ◽

Jian Xi Fan ◽

Lan Tao You ◽

Yan Wang

Keyword(s):

Parallel Computing ◽

Binary Tree ◽

Interconnection Network ◽

Binary Trees ◽

Complete Binary Tree ◽

Locally Twisted Cubes ◽

Twisted Cube ◽

Twisted Cubes

The locally twisted cube is a newly introduced interconnection network for parallel computing, which possesses many desirable properties. In this paper, the problem of embedding complete binary trees into locally twisted cubes is studied.Let LTQn(V;E) denote the n-dimensional locally twisted cube.We find the following result in this paper: for any integern ≥ 2,we show that a complete binary tree with 2n—1 nodes can be embedded into the LTQn with dilation 2.

Download Full-text

OPTIMUM CONGESTED ROUTING STRATEGY ON TWISTED CUBES

Journal of Interconnection Networks ◽

10.1142/s0219265900000081 ◽

2000 ◽

Vol 01 (02) ◽

pp. 115-134 ◽

Cited By ~ 3

Author(s):

TSENG-KUEI LI ◽

JIMMY J. M. TAN ◽

LIH-HSING HSU ◽

TING-YI SUNG

Keyword(s):

Shortest Path ◽

Time Complexity ◽

Interconnection Network ◽

Routing Algorithm ◽

Optimum Time ◽

Shortest Path Routing ◽

Routing Strategy ◽

Bisection Width ◽

Twisted Cube ◽

Twisted Cubes

Given a shortest path routing algorithm of an interconnection network, the edge congestion is one of the important factors to evaluate the performance of this algorithm. In this paper, we consider the twisted cube, a variation of the hypercube with some better properties, and review the existing shortest path routing algorithm8. We find that its edge congestion under the routing algorithm is high. Then, we propose a new shortest path routing algorithm and show that our algorithm has optimum time complexity O(n) and optimum edge congestion 2n. Moreover, we calculate the bisection width of the twisted cube of dimension n.

Download Full-text

The Star-Structure Connectivity and Star-Substructure Connectivity of Hypercubes and Folded Hypercubes

The Computer Journal ◽

10.1093/comjnl/bxab133 ◽

2021 ◽

Author(s):

Lina Ba ◽

Heping Zhang

Keyword(s):

Fault Tolerance ◽

Open Problem ◽

Connected Subgraph ◽

Vertex Connectivity ◽

Minimum Cardinality ◽

Connected Graphs ◽

Folded Hypercube ◽

Star Structure

Abstract As a generalization of vertex connectivity, for connected graphs $G$ and $T$, the $T$-structure connectivity $\kappa (G; T)$ (resp. $T$-substructure connectivity $\kappa ^{s}(G; T)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $T$ (resp. to a connected subgraph of $T$) so that $G-F$ is disconnected. For $n$-dimensional hypercube $Q_{n}$, Lin et al. showed $\kappa (Q_{n};K_{1,1})=\kappa ^{s}(Q_{n};K_{1,1})=n-1$ and $\kappa (Q_{n};K_{1,r})=\kappa ^{s}(Q_{n};K_{1,r})=\lceil \frac{n}{2}\rceil $ for $2\leq r\leq 3$ and $n\geq 3$ (Lin, C.-K., Zhang, L.-L., Fan, J.-X. and Wang, D.-J. (2016) Structure connectivity and substructure connectivity of hypercubes. Theor. Comput. Sci., 634, 97–107). Sabir et al. obtained that $\kappa (Q_{n};K_{1,4})=\kappa ^{s}(Q_{n};K_{1,4})= \lceil \frac{n}{2}\rceil $ for $n\geq 6$ and for $n$-dimensional folded hypercube $FQ_{n}$, $\kappa (FQ_{n};K_{1,1})=\kappa ^{s}(FQ_{n};K_{1,1})=n$, $\kappa (FQ_{n};K_{1,r})=\kappa ^{s}(FQ_{n};K_{1,r})= \lceil \frac{n+1}{2}\rceil $ with $2\leq r\leq 3$ and $n\geq 7$ (Sabir, E. and Meng, J.(2018) Structure fault tolerance of hypercubes and folded hypercubes. Theor. Comput. Sci., 711, 44–55). They proposed an open problem of determining $K_{1,r}$-structure connectivity of $Q_n$ and $FQ_n$ for general $r$. In this paper, we obtain that for each integer $r\geq 2$, $\kappa (Q_{n};K_{1,r})$ $=\kappa ^{s}(Q_{n};K_{1,r})$ $=\lceil \frac{n}{2}\rceil $ and $\kappa (FQ_{n};K_{1,r})=\kappa ^{s}(FQ_{n};K_{1,r})= \lceil \frac{n+1}{2}\rceil $ for all integers $n$ larger than $r$ in quare scale. For $4\leq r\leq 6$, we separately confirm the above result holds for $Q_n$ in the remaining cases.

Download Full-text

Structure Fault Tolerance of Recursive Interconnection Networks

The Computer Journal ◽

10.1093/comjnl/bxz139 ◽

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.

Download Full-text

Structure Fault Tolerance of Recursive Interconnection Networks

The Computer Journal ◽

10.1093/comjnl/bxz159 ◽

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 $\mathcal{F}=\{T_{1},T_{2},...,T_{m}\}$ (resp. $\mathcal{F}=\{T_{1}^{^{\prime}},T_{2}^{^{\prime}},...,T_{m}^{^{\prime}}\}$) 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 results in Lin, C.-K., Zhang, L., Fan, J. and Wang, D. (2016, Structure connectivity and substructure connectivity of hypercubes. Theor. Comput. Sci., 634, 97–107) and Lv, Y., Fan, J., Hsu, D.F. and Lin, C.-K. (2018, Structure connectivity and substructure connectivity of $k$-ary $n$-cubes. Inf. Sci., 433, 115–124).

Download Full-text

Hamiltonian Cycles Passing Through Prescribed Edges in Locally Twisted Cubes

Journal of Interconnection Networks ◽

10.1142/s0219265921500201 ◽

2021 ◽

Author(s):

Dongqin Cheng

Keyword(s):

Hamiltonian Cycle ◽

Disjoint Paths ◽

Hamiltonian Cycles ◽

Induced Subgraph ◽

Locally Twisted Cubes ◽

Twisted Cube ◽

Vertex Disjoint ◽

Cube Formula ◽

Twisted Cubes

Let [Formula: see text] be a set of edges whose induced subgraph consists of vertex-disjoint paths in an [Formula: see text]-dimensional locally twisted cube [Formula: see text]. In this paper, we prove that if [Formula: see text] contains at most [Formula: see text] edges, then [Formula: see text] contains a Hamiltonian cycle passing through every edge of [Formula: see text], where [Formula: see text]. [Formula: see text] has a Hamiltonian cycle passing through at most one prescribed edge.

Download Full-text