Hypercube-like Networks Research Articles

Connectivity and super connectivity of folded hypercube-like networks

We define a new class of graphs by generalizing the hypercube-like networks to folded hypercube-like networks FHn. We also investigate the reliability of this class of graphs in terms of the (edge) connectivity and super (edge) connectivity. Since this class contains two well-known hypercube variants, namely the folded hypercube and the folded crossed cube, this paper extends some previously known results.

Two-disjoint-cycle-cover vertex bipancyclicity of bipartite hypercube-like networks

Let r1,r2 be two integers such that r2≥r1≥0. A bipartite graph G is two-disjoint-cycle-cover vertex [r1,r2]-bipancyclic (2-DCC vertex [r1,r2]-bipancyclic for short) if for any two vertices u,v∈V(G) and any even integer ℓ satisfying r1≤ℓ≤r2, there exist two vertex-disjoint cycles C1 and C2 in G with |V(C1)|=ℓ and |V(C2)|=|V(G)|−ℓ such that u∈V(C1) and v∈V(C2). In this paper, we study the 2-DCC vertex bipancyclicity of the n-dimensional bipartite hypercube-like network, which is one class of hypercube-generalized networks. As a consequence, we show that an n-dimensional bipartite hypercube-like network is 2-DCC vertex [4,2n−1]-bipancyclic for n≥3. In particular, it provides an application that n-dimensional hypercube and bicube are also 2-DCC vertex [4,2n−1]-bipancyclic for n≥3.

On the g-component connectivity of hypercube-like networks

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. Let be an integer and G be a connected graph. A g-component cut of G is a vertex set S such that G−S has at least g components. The g-component connectivity of G is the size of the smallest g-component cut. Determining the g-component connectivity is still an unsolved problem in many interconnection networks. In this paper, we prove the lower bound of the g-component connectivity of any n-dimensional hypercube-like networks. We also determine the g-component connectivity of varietal hypercubes and crossed cubes which are the members of hypercube-like networks. As a by-product, we characterize the optimal g-component cut under the condition that any two vertices have exactly two common neighbors if they have of any n-dimensional hypercube-like networks.

The generalized 4-connectivity of folded hypercube

ABSTRACT In recent years, a growing number of concepts have been proposed to better assess network fault tolerance, among which the generalized -connectivity has been widely used in the research of fault tolerance of interconnection networks. For connected graph , refers to the maximum number of internally disjoint trees in to connect , where with . The generalized -connectivity of G . The -dimensional folded hypercube , as a hypercube-like network, is obtained by adding edges on -dimensional hypercube . In this paper, we discuss the generalized 4-connectivity of and show that for .

A Local Diagnosis Algorithm for Hypercube-like Networks under the BGM Diagnosis Model

System diagnosis is process of identifying faulty nodes in a system. An efficient diagnosis is crucial for a multiprocessor system. The BGM diagnosis model is a modification of the PMC diagnosis model, which is a test-based diagnosis. In this paper, we present a specific structure and propose an algorithm for diagnosing a node in a system under the BGM model. We also give a polynomial-time algorithm that a node in a hypercube-like network can be diagnosed correctly in three test rounds under the BGM diagnosis model.

The unpaired many-to-many k-disjoint paths in bipartite hypercube-like networks

In this paper, we discuss about the problem of unpaired many-to-many disjoint paths in bipartite hypercube-like networks with faulty elements. Let F be a faulty set with f faulty elements in an n-dimensional bipartite hypercube-like network G. For every positive integer k≥1 satisfying f+k≤n−1, and two arbitrary sets S and T with k fault-free vertices in distinct parts of the bipartition of G, there exists an unpaired fault-free k-disjoint path collection joining S and T in G−F, which contains at least 2n−2fv vertices, where fv is the number of faulty vertices in G. This result is optimal in the worst case, and the bound f+k≤n−1 is tight.

Component edge connectivity of hypercube-like networks

As a generalization of the traditional connectivity, the g-component edge connectivity cλg(G) of a non-complete graph G is the minimum number of edges to be deleted from the graph G such that the resulting graph has at least g components. Hypercube-like networks (HL-networks for short) are obtained by manipulating some pairs of edges in hypercubes, which contain several famous interconnection networks such as twisted cubes, Möbius cubes, crossed cubes, locally twisted cubes. In this paper, we determine the (g+1)-component edge connectivity of the n-dimensional HL-networks for g≤2⌈n2⌉, n≥8.

Edge-fault-tolerant strong Menger edge connectivity on regular graphs

The connectivity of a graph is an important issue in graph theory and is also one of the most important factors in evaluating the reliability and fault tolerance of a network. A graph G is called m-edge-fault-tolerant strongly Menger (m-EFTSM for short) edge connected if there are min⁡{degG−F⁡(x),degG−F⁡(y)} edge-disjoint paths between any two different vertices x and y in G−F for any F⊆E(G) with |F|≤m.In this paper, we give a necessary and sufficient condition of EFTSM edge connectivity on regular graphs. And we obtain several optimal results about EFTSM edge connectivity on (1, 2)-matching composition networks, each of which is constructed by connecting two graphs via one or two perfect matchings. As applications, we show that the class of n-dimensional hypercube-like networks (included hypercube, crossed cube et al.) are (n−2)-EFTSM edge connected; show that the n-dimensional folded hypercube is (n−1)-EFTSM edge connected, and show that the n-dimensional augmented cube is (2n−3)-EFTSM edge connected. The bounds (n−2),(n−1) and (2n−3) are sharp.

The bipanconnectivity of bipartite hypercube-like networks

Bipanconnectivity is an important parameter in bipartite networks related on the embedding problem of linear arrays and rings. In this paper, we study the fault-tolerant bipanconnectivity of bipartite n-dimensional hypercube-like networks, denoted as Bn′. We show that for any n-dimensional bipartite hypercube-like network G∈Bn′ with f faulty elements (edges and/or vertices), including fv faulty vertices such that f≤n−2, for each pair of fault-free vertices of distance d in G, there exists a fault-free path of length l linking them, where 2n−4≤l≤2n−2fv−1 and l−d≡0 (mod 2). Our result is optimal when considering the number of faulty elements.

Structure and Substructure Connectivity of Hypercube-Like Networks

The connectivity of a graph [Formula: see text], [Formula: see text], is the minimum number of vertices whose removal disconnects [Formula: see text], and the value of [Formula: see text] can be determined using Menger’s theorem. It has long been one of the most important factors that characterize both graph reliability and fault tolerability. Two extensions to the classic notion of connectivity were introduced recently: structure connectivity and substructure connectivity. Let [Formula: see text] be isomorphic to any connected subgraph of [Formula: see text]. The [Formula: see text]-structure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] such that every element of [Formula: see text] is isomorphic to [Formula: see text], and the removal of [Formula: see text] disconnects [Formula: see text]. The [Formula: see text]-substructure connectivity of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum set [Formula: see text] of connected subgraphs in [Formula: see text] whose removal disconnects [Formula: see text] and every element of [Formula: see text] is isomorphic to a connected subgraph of [Formula: see text]. The family of hypercube-like networks includes many well-defined network architectures, such as hypercubes, crossed cubes, twisted cubes, and so on. In this paper, both the structure and substructure connectivity of hypercube-like networks are studied with respect to the [Formula: see text]-star [Formula: see text] structure, [Formula: see text], and the [Formula: see text]-cycle [Formula: see text] structure. Moreover, we consider the relationships between these parameters and other concepts.

Conditional diagnosability of component-composition graphs under the PMC model

Diagnosability is an important metric for measuring the reliability of multiprocessor systems. The conditional diagnosability, which is more general than the classical diagnosability, is to measure the diagnosability of a multiprocessor system under the assumption that all of the neighbors of any node in the system cannot fail at the same time. This paper adopts the PMC model and outlines the common properties of a wide class of interconnection networks, called component-composition graphs (CCGs), to determine their conditional diagnosability by using the derived properties. As applications, the diagnosability of hypercube-like networks (including hypercubes, crossed cubes, Möbius cubes, twisted cubes, locally twisted cubes, generalized twisted cubes, and recursive circulants), and bubble-sort graphs, all of which belong to CCGs can be readily obtained.

Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks (THLNs)

It is known widely that an interconnection network can be denoted by a graph G = ( V , E ) , where V denotes the vertex set and E denotes the edge set. Investigating structures of G is necessary to design a suitable topological structure of interconnection network. One of the critical issues in evaluating an interconnection network is graph embedding, which concerns whether a host graph contains a guest graph as its subgraph. Linear arrays (i.e., paths) and rings (i.e., cycles) are two ordinary guest graphs (or basic networks) for parallel and distributed computation. In the process of large-scale interconnection network operation, it is inevitable that various errors may occur at nodes and edges. It is significant to find an embedding of a guest graph into a host graph where all faulty nodes and edges have been removed. This is named as fault-tolerant embedding. The twisted hypercube-like networks ( T H L N s ) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of n-dimensional (n-D) T H L N s . Let G n be an n-D T H L N and F be a subset of V ( G n ) ∪ E ( G n ) with | F | ≤ n - 2 . We show that for two different arbitrary correct vertices u and v, there is a faultless path P u v of every length l with 2 n - 1 - 1 ≤ l ≤ 2 n - f v - 1 - α , where α = 0 if vertices u and v form a normal vertex-pair and α = 1 if vertices u and v form a weak vertex-pair in G n - F ( n ≥ 5 ).

Fault-tolerant embedding of starlike trees into restricted hypercube-like graphs

A d-starlike tree (or a d-quasistar) is a subdivision of a star tree of degree d. A family of hypercube-like interconnection networks, called restricted hypercube-like graphs, includes most non-bipartite hypercube-like networks found in the literature such as twisted cubes, crossed cubes, Möbius cubes, recursive circulant G(2m,4) of odd m, etc. In this paper, we prove that given an arbitrary fault-free vertex r in an m-dimensional restricted hypercube-like graph with a set F of faults (vertex and/or edge faults) and d positive integers, l1,l2,…,ld, whose sum is equal to the number of fault-free vertices minus one, there exists a d-starlike tree rooted at r, each of whose subtrees forms a fault-free path on li vertices for i∈{1,2,…,d}, provided |F|≤m−2 and |F|+d≤m. The bounds on |F| and |F|+d are the maximum possible.

Edge-fault-tolerant strong Menger edge connectivity on the class of hypercube-like networks

From Menger’s theorem, a graph is k-connected if and only if there are at least k-internally disjoint paths between any two distinct vertices. Therefore, the number of internally disjoint paths between two vertices may be larger than the connectivity. Motivated by this observation, Oh and Chen (resp., Qiao and Yang) proposed the (fault-tolerant) strong Menger (resp., edge) connectivity as follows. A connected graph G is called strongly Menger (edge) connected if for any two distinct vertices x,y in G, there are min{degG(x),degG(y)}(-edge)-disjoint paths between x and y. A graph G is called m(-edge)-fault-tolerant strongly Menger (edge) connected if G−F remains strongly Menger (edge) connected for an arbitrary set F⊆V(G) (resp., F⊆E(G)) with |F|≤m. A graph G is called m-conditional (edge)-fault-tolerant strongly Menger (edge) connected if G−F remains strongly Menger (edge) connected for an arbitrary set F⊆V(G) (resp., F⊆E(G)), |F|≤m and δ(G−F)≥2. In this paper, we show that all n-dimensional hypercube-like networks are (n−2)-edge-fault-tolerant strongly Menger edge connected and (3n−8)-conditional edge-fault-tolerant strongly Menger edge connected for n≥3 which generalizes the results of Qiao and Yang in 2017. Our results are all optimal with respect to the maximum number of tolerated edge faults.

The pessimistic diagnosability of graphs and its applications to four kinds of interconnection networks

ABSTRACTIn a simple graph , let be the minimum cardinality of the neighbourhoods of any two adjacent vertices, i.e. . Let be the connectivity of G. In this paper, we prove that the pessimistic diagnosability of G, denoted by , is equal to if the following two conditions hold: (1) for any subset with , ; (2) . As examples of its applications, we prove that the pessimistic diagnosabilities of n-dimensional hypercube-like network , dual-cube , pancake network , and burnt pancake graph are (), (), () and (), respectively.

Fault-Tolerant Hamiltonian Connectivity of Twisted Hypercube-Like Networks <italic>THLNs</italic>

The twisted hypercube-like networks (THLNs) include some well-known hypercube variants. A graph $G$ is $k$ -fault-tolerant Hamiltonian connected if $G-F$ remains Hamiltonian connected for every $F\subset V(G)\cup E(G)$ with $|F|\leq k$ . This paper is concerned with the fault-tolerant Hamiltonian connectivity of an $n$ -dimensional ( $n$ -D) THLN. Let $G_{n}$ be an $n$ -D THLN ( $n\geq 5$ ) and $F$ be a subset of $V(G_{n})\cup E(G_{n})$ with $|F|\leq n-2$ . We show that for arbitrary vertex-pair $(u,v)$ in $G_{n}-F$ , there exists a $(n-2)$ -fault-tolerant Hamiltonian path joining vertices $u$ and $v$ except $(u,v)$ being a weak vertex-pair in $G_{n}-F$ . The technical theorem proposed in this paper can be applied to several multiprocessor systems, including $n$ -D crossed cubes $CQ_{n}$ , $n$ -D twisted cubes $TQ_{n}$ for odd $n$ , $n$ -D locally twisted cubes $LTQ_{n}$ , and $n$ -D Mobius cubes $MQ_{n}$ .

On g-extra connectivity of hypercube-like networks

Given a connected graph G and a non-negative integer g, the g-extra connectivityκg(G) of G is the minimum cardinality of a set of vertices in G, if it exists, whose deletion disconnects G and leaves each remaining component with more than g vertices. This paper focuses on the g-extra connectivity of hypercube-like networks (HL-networks for short). All the known results suggest the equality κg(Xn)=fn(g) holds, where Xn is an n-dimensional HL-network, fn(g)=n(g+1)−g(g+3)2, n≥5 and 0≤g≤n−3. However, in this paper, we show that this equality does not hold in general. We also prove that κg(Xn)≥fn(g) holds for n≥5 and 0≤g≤n−3. This enables us to give a sufficient condition for the equality κg(Xn)=fn(g), which is then used to determine the g-extra connectivity of HL-networks for some small g or the g-extra connectivity of some particular subfamily of HL-networks.

Conditional edge-fault hamiltonian-connectivity of restricted hypercube-like networks

A graph G is considered conditional k-edge-fault hamiltonian-connected if, after k faulty edges are removed from G, under the assumption that each node is incident to at least three fault-free edge...

Conditional edge-fault hamiltonian-connectivity of restricted hypercube-like networks

A graph G is considered conditional k-edge-fault hamiltonian-connected if, after k faulty edges are removed from G, under the assumption that each node is incident to at least three fault-free edges, a hamiltonian path exists between any two distinct nodes in the resulting graph. This paper focuses on the conditional edge-fault hamiltonian-connectivity of a wide class of interconnection networks called restricted hypercube-like networks (RHLs). An n-dimensional RHL (RHLn) is proved to be conditional (2n−7)-edge-fault hamiltonian-connected for n≥5. The technical theorem proposed in this paper is then applied to show that several multiprocessor systems, including n-dimensional crossed cubes, n-dimensional twisted cubes for odd n, n-dimensional locally twisted cubes, n-dimensional generalized twisted cubes, n-dimensional Möbius cubes, and recursive circulants G(2n,4) for odd n, are all conditional (2n−7)-edge-fault hamiltonian-connected.

Improved Precise Fault Diagnosis Algorithm for Hypercube-Like Systems Based on the Comparison Diagnosis Model

Multiprocessor systems are being increasingly adopted and the system reliability is an important perspective for multiprocessor systems. The fault diagnosis has become crucial for achieving high reliability in multiprocessor systems. The precise fault diagnosis diagnoses all processors correctly. In the comparison-based model, it allows a processor to perform diagnosis by contrasting the responses from a pair of neighboring processors through sending the identical assignment. On the basis of comparison-based model, Sengupta and Dahbura (“On self-diagnosable multiprocessor systems: diagnosis by the comparison approach,” IEEE Transaction on Computers, vol. 41, no. 11, pp. 1386–1396, 1992) put forward the MM* model, any processor c diagnoses two processors c1 and c2 if c has direct communication links to them. Sengupta and Dahbura also designed an O(N5)-time precise fault diagnosis algorithm to diagnose faulty processors for general topologies by using the MM* model, where N is the cardinality of processor set in multiprocessor systems. Lately, Ye and Hsieh (“A scalable comparison-based diagnosis algorithm for hypercube-like net-works,” IEEE Transaction on Reliability, vol. 62, no. 4, pp. 789–799, 2013) devised an precise fault diagnosis algorithm to diagnose all faulty processors for hypercube-like networks by using the MM* model with O(N(log2N)2) time complexity. On the basis of Hamiltonian cycle properties, we improve the aforementioned results by presenting an O(N)-time precise fault diagnosis algorithm to diagnose all faulty processors for hypercube-like networks by using the MM* model.