G-extra Connectivity Research Articles

Reliability Analysis of (n, k)-Bubble-Sort Networks Based on Extra Conditional Fault

Given a graph G=(V(G),V(E)), a non-negative integer g and a set of faulty vertices F⊆V(G), the g-extra connectivity of G, denoted by κg(G), is the smallest cardinality of F, whose value of deletion, if exists, will disconnect G and give each remaining component at least g+1 vertices. The g-extra diagnosability of the graph G, denoted by tg(G), is the maximum cardinality of the set F of fault vertices that the graph can guarantee to identify under the condition that each fault-free component has more than g vertices. In this paper, we determine that g-extra connectivity of (n,k)-bubble-sort network Bn,k is κg(Bn,k)=n+g(k−2)−1 for 4≤k≤n−1 and 0≤g≤n−k. Afterwards, we show that g-extra diagnosability of Bn,k under the PMC model (4≤k≤n−1 and 0≤g≤n−k) and MM* model (4≤k≤n−1 and 0≤g≤min{n−k−1,k−2}) is tg(Bn,k)=n+g(k−1)−1, respectively.

High fault-tolerant performance of the divide-and-swap cube network

Two crucial metrics used to evaluate the fault tolerance of interconnection networks are connectivity and diagnosability. By improving the connectivity and diagnosability of the interconnection network, its fault tolerance can be enhanced. In this paper, we focus on determining the g-extra connectivity (0≤g≤10) of the divide-and-swap cube DSCn, as well as its diagnosability based on the pessimistic diagnosis strategy and g-extra precise diagnosis strategy, under the PMC and MM* models. The research analysis suggests that compared with some other connectivity and diagnosability of DSCn, such as classical connectivity, structure connectivity, super connectivity, and classical diagnosability, the extra connectivity/diagnosability and pessimistic diagnosability of DSCn enable it to have a higher fault tolerance. Moreover, we propose two O(Nlog2⁡N) effective diagnosis algorithms of DSCn: the g-extra diagnosis algorithm (EX-DiagnosisDSCn) and the pessimistic diagnosis algorithm (PE-DiagnosisDSCn), where the EX-DiagnosisDSCn algorithm can accurately diagnose the state of all processors in DSCn.

Reliability evaluation for a class of recursive match networks

Due to system failures in endlessly, efforts are always strived to improve the reliability of the system. As a fundamental problem in large-scale parallel and distributed systems, connectivity and diagnosability, especially g-extra connectivity and g-extra conditional diagnosability, have been widely followed. This work proposes a new class of recursive networks including the BCube—conditional recursive match networks (CRMNs). To explore the reliability of CRMNs, we get the g-extra connectivity and g-extra conditional diagnosability of CRMNs under the PMC model and the MM* model for 0≤g≤3 and g=2m+1−1(m≥0). And these results can not only be applied to the BCube, but also be used to directly obtain the corresponding g-extra connectivity and g-extra conditional diagnosability of some other networks except the BCube. In addition, we put forward the conjectures as regards the g-extra connectivity and g-extra conditional diagnosability of CRMNs with g>0.

Generalized fault-tolerance for enhanced hypercubes

Fault-tolerance is an important indicator to measure the stability of the interconnection network. The connectivity can reflect the reliability of a network. The g-extra connectivity κg(G) of a network G is a generalization of the connectivity. The enhanced hypercube Qn,k is an important variant of hypercube Qn which has been received many attentions in recent years. The g-extra connectivity of the enhanced hypercube was investigated for some special g and k in the literature. We fill the gap of k and determine κg(Qn,k) for the remaining cases of k when g≤3. In summary, we show that κg(Qn,3)=(g+1)n−4(g−1) and κg(Qn,4)=(g+1)n−g(g−1) where n≥8 and 1≤g≤3.

The regular edge connectivity of regular networks

Classical connectivity is a vital metric to assess the reliability of interconnection networks, while it has defects in the defective assumption that all neighbours of one node may fail concurrently. To overcome this deficiency, many new generalizations of traditional connectivity, such as g-component (edge) connectivity, restricted (edge) connectivity, g-extra (edge) connectivity and so on, have been suggested. For any positive integer a, the a-regular edge connectivity of a connected graph G is the minimum cardinality of an edge cut, whose deletion disconnects G so that each component is a a-regular graph. In this work, we investigate the a-regular edge connectivity of some regular networks, which provides a novel idea for further exploring the reliability of networks. Finally, simulations are carried out to compare the a-regular connectivity with other types of connectivity in some regular networks. The comparison results imply that the a-regular edge connectivity has its superiority.

Generalized Connectivity of the Mycielskian Graph under g-Extra Restriction

The g-extra connectivity is a very important index to evaluate the fault tolerance, reliability of interconnection networks. Let g be a non-negative integer, G be a connected graph with vertex set V and edge set E, a subset S⊆V is called a g-extra cut of G if the graph induced by the set G−S is disconnected and each component of G−S has at least g+1 vertices. The g-extra connectivity of G, denoted as κg(G), is the cardinality of the minimum g-extra cut of G. Mycielski introduced a graph transformation to discover chromatic numbers of triangle-free graphs that can be arbitrarily large. This transformation converts a graph G into a new compound graph called μ(G), also known as the Mycielskian graph of G. In this paper, we study the relationship on g-extra connectivity between the Mycielskian graph μ(G) and the graph G. In addition, we show that κ3(μ(G))=2κ1(G)+1 for κ1(G)≤min{4,⌊n2⌋}, and prove the bounds of κ2g+1(μ(G)) for g≥2.

On the g-extra connectivity of augmented cubes

A g-extra cut of a non-complete graph G,g≥0, is a set of vertices in G whose removal disconnects the graph, while every component in the survival graph contains at least g+1 vertices. The g-extra connectivity of G then refers to the size of a minimum g-extra cut of G.The augmented hypercube, denoted by AQn,n≥3, is a rich variant of the hypercube structure. In this paper, we present a sequence of construction based upper bounds of its g-extra connectivity, study its lower bound via the super connectedness property, and suggest an asymptotically tight bound.

Structure connectivity of data center networks

Last decade, numerous giant data center networks are built to provide increasingly fashionable web applications. For two integers m≥0 and n≥2, the m-dimensional DCell network with n-port switches Dm,n and n-dimensional BCDC network Bn have been proposed. As a generalization of connectivity, structure (substructure) connectivity was recently proposed. Let G and H be two connected graphs. Let F be a set whose elements are subgraphs of G, and every member of F is isomorphic to H (resp. a connected subgraph of H). Then H-structure connectivity κ(G;H) (resp. H-substructure connectivity κs(G;H)) of G is the size of a smallest set of F such that G−F is disconnected or the singleton. Then it is meaningful to calculate the structure connectivity of data center networks on some common structures, such as star K1,t, path Pk, cycle Ck, complete graph Ks and so on. In this paper, we obtain that κ(Dm,n;K1,t)=κs(Dm,n;K1,t)=⌈n−11+t⌉+m for 1≤t≤m+n−2 and κ(Dm,n;Ks)=⌈n−1s⌉+m for 3≤s≤n−1 by analyzing the structural properties of Dm,n. We also compute κ(Bn;H) and κs(Bn;H) for H∈{K1,t,Pk,Ck|1≤t≤2n−3,6≤k≤2n−1} and n≥5 by using g-extra connectivity of Bn.

The t/k-Diagnosability and a t/k Diagnosis Algorithm of the Data Center Network BCCC under the MM* Model

The evaluation of the fault diagnosis capability of a data center network (DCN) is important research in measuring network reliability. The g-extra diagnosability is defined under the condition that every component except the fault vertex set contains at least g+1 vertices. The t/k diagnosis strategy is that the number of fault nodes does not exceed t, and all fault nodes can be isolated into a set containing up to k fault-free nodes. As an important data center network, BCube Connected Crossbars (BCCC) has many excellent properties that have been widely studied. In this paper, we first determine that the g-extra connectivity of BCn,k for 0≤g≤n−1. Based on this, we establish the g-extra conditional diagnosability of BCn,k under the MM* model for 1≤g≤n−1. Next, based on the conclusion of the largest connected component in g-extra connectivity, we prove that the t/k-diagnosability of BCn,k under the MM* model for 1≤k≤n−1. Finally, we present a t/k diagnosis algorithm on BCCC under the MM* model. The algorithm can correctly identify all nodes at most k nodes undiagnosed. So far, t/k-diagnosability and diagnosis algorithms for most networks in the MM* model have not been studied.

The g-Extra Connectivity of the Strong Product of Paths and Cycles

Let G be a connected graph and g be a non-negative integer. A vertex set S of graph G is called a g-extra cut if G−S is disconnected and each component of G−S has at least g+1 vertices. The g-extra connectivity of G is the minimum cardinality of a g-extra cut of G if G has at least one g-extra cut. For two graphs G1=(V1,E1) and G2=(V2,E2), the strong product G1⊠G2 is defined as follows: its vertex set is V1×V2 and its edge set is {(x1,x2)(y1,y2)|x1=x2 and y1y2∈E2; or y1=y2 and x1x2∈E1; or x1x2∈E1 and y1y2∈E2}, where (x1,x2),(y1,y2)∈V1×V2. In this paper, we obtain the g-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles.

The g-extra connectivity of folded crossed cubes

There are various ways to measure the reliability and fault tolerance of diverse networks. The g-extra connectivity κg(G) of a connected graph G is the minimal cardinality of vertex set F, if any, whose deletion disconnects G and every remaining component of G−F has at least g+1 vertices. Folded crossed cubes FCQn, a kind of interconnection network, has more reliable properties. In this paper, we explore the g-extra connectivity of n-dimensional folded crossed cubes. It is shown that when 0≤g≤⌊n2⌋, κg(FCQn)=(g+1)n−g−(g2)+1 for n≥8. As a byproduct, we get g-extra conditional fault-diagnosability tgp˜(FCQn)=(g+1)n−(g2)+1 of FCQn under PMC model.

Reliability of the round matching composition networks based on g-extra conditional fault

Connectivity and diagnosability are very important parameters for the tolerant of an interconnection network. Let F be a vertex set of graph G, the g-extra connectivity k˜(g)(G) of G requires each component of G−F has at least (g+1) vertices. The g-extra conditional diagnosability t˜(g)(G) is the maximum number of faulty vertices that the graph can be guaranteed to identify under the condition that each fault-free component has at least g+1 vertices. In this paper, we study the problem about the g-extra connectivity k˜(g)(G) of the round matching composition networks, and obtain some results about the g-extra conditional diagnosability of the round matching composition networks under the PMC model and MM* model, respectively.

On the g-Extra Connectivity of the Enhanced Hypercubes

Abstract Reliability evaluation of interconnection networks is of significant importance to the design and maintenance of interconnection networks. The extra connectivity is an important parameter for the reliability evaluation of interconnection networks and is a generalization of the traditional connectivity. Let $g\geq 0$ be an integer and $G$ be a connected graph; the $g$-extra connectivity of $G$ is the minimum cardinality of a set of vertices in $G$, if it exists, whose removal disconnects $G$ and leaves every component with more than $g$ vertices. Determining the $g$-extra connectivity is still an unsolved problem in many interconnection networks. Let $n$, $k$ be positive integers. Let $Q_{n,k}\, (1 \leq k \leq n-1)$ denote the $(n, k)$-enhanced hypercube. In this paper, we determine the $g$-extra connectivity of $Q_{n,k}$ is $(n+1)(g+1)-\frac{g(g+3)}{2}$ for $0\leq g\leq n-k-1$, $4\leq k\leq n-5\,(n\geq 9)$. Some previous results in [Zhang, M. and Zhou, J. (2015) On g-extra connectivity of folded hypercubes. Theor. Comput. Sci., 593, 146–153.] and [Sabir, E., Mamut, A. and Vumar, E. (2019) The extra connectivity of the enhanced hypercubes. Theor. Comput. Sci., 799, 22–31.] are extended.

Reliability of DQcube Based on g-Extra Conditional Fault

Abstract Diagnosability and connectivity are important metrics for the reliability and fault diagnosis capability of interconnection networks, respectively. The g-extra connectivity of a graph G, denoted by $\kappa _g(G)$, is the minimum number of vertices whose deletion will disconnect the network and every remaining component has more than $g$ vertices. The g-extra conditional diagnosability of graph G, denoted by $t_g(G)$, is the maximum number of faulty vertices that the graph G can guarantee to identify under the condition that every fault-free component contains at least g+1 vertices. In this paper, we first determine that g-extra connectivity of DQcube is $\kappa _g(G)=(g+1)(n+1)-\frac{g(g+3)}{2}$ for $0\leq g\leq n-3$ and then show that the g-extra conditional diagnosability of DQcube under the PMC model $(n\geq 4, 1\leq g\leq n-3)$ and the MM$^\ast$ model $(n\geq 7, 1\leq g\leq \frac{n-3}{4})$ is $t_g(G)=(g+1)(n+1)-\frac{g(g+3)}{2}+g$, respectively.

The 2-Extra Connectivity of Wheel Networks

Connectivity is a significant metric for evaluating the error lenience of an interconnected net G=V,E. In the paper (Fàbrega and Fiol, 1996), the authors addressed the g-extra connectivity which is applied to better measure the consistency and error tolerance of G. As a special Cayley graph, the n-dimensional wheel network CWn has some desirable features. This paper shows that the g-extra connectivity of CWnn≥6 is 6n−12 when g=2.

The g-extra diagnosability of the generalized exchanged hypercube

Diagnosability of a self-diagnosable interconnection structure specifies the maximum number of faulty vertices such a structure can identify by itself. A variety of diagnosability models have been suggested. It turns out that a diagnosability property of a network structure is closely associated with its relevant connectivity property. Based on this observation, a general diagnosability derivation process has been suggested. The g-extra connectivity of a graph G characterizes the size of a minimum vertex set F such that, when it is removed, every component in the disconnected survival graph, contains at least g + 1 vertices. In this paper, we discuss the aforementioned general derivation process, derive the g-extra connectivity, and then apply the aforementioned general process to reveal the g-extra diagnosability of the generalized exchanged hypercube.

The Relationship Between Extra Connectivity and t/k-Diagnosability of Regular Networks

The g-extra connectivity of a multiprocessor system modeled by a graph G, denoted by [Formula: see text] (G), is the minimum number of removed vertices such that the network is disconnected and each residual component has no less than g + 1 vertices. The t/k-diagnosis strategy can detect up to t faulty processors which might include at most k misdiagnosed processors. These two parameters are important to measure the fault tolerant ability of a multiprocessor system. The extra connectivity and t/k-diagnosability of many well-known networks have been investigated extensively and independently. However, the general relationship between the extra connectivity and the t/k-diagnosability of general regular networks has not been established. In this paper, we explore the relationship between the k-extra connectivity and t/k-diagnosability for regular networks under the classic PMC diagnostic model. More specifically, we derive the relationship between 1-extra connectivity and pessimistic diagnosability for regular networks. Furthermore, the t/k-diagnosability and pessimistic diagnosability of some networks, including star network, BC networks, Cayley graphs generated by transposition trees etc., are determined.

Reliability and conditional diagnosability of hyper bijective connection networks

The g-extra connectivity and diagonalisability are two important metrics to fault-tolerance and robustness of a multiprocessor system whose network structure is modelled by a graph. In this work, we explore the reliability of a newly proposed network called hyper bijective connection networks (HBC, for short), which is an extension of the family of the well-known interconnection networks, such as hypercube and its variants. We prove that 2-extra vertex connectivity and 3-extra vertex connectivity of n-dimensional HBC are 3n + m−6 for and and 4n + m−8 for and , respectively. Using its desirable fault-tolerance, we show that the conditional diagonalizabilities of n-dimensional HBC under the PMC model are m + 4n−7 (resp., 4n−5) for and (resp., m = 3 and ) and its conditional diagonalizability under MM model is m + 3n−6 for and .

The extra connectivity of the enhanced hypercubes

As one of the best parameters to evaluate the fault tolerance of interconnection networks, extra connectivity plays an important role in multiprocessor systems for reliable computing. Given a graph G and a non-negative integer g, the g-extra connectivity (resp. g-extra edge connectivity) of G, denoted by κg(G) (resp. λg(G)), is the minimum cardinality of a set of vertices (resp. edges) in G, if exists, whose deletion disconnects G and leaves each remaining component with at least g+1 vertices. In this paper, we determine κg(Qn,k) and λg(Qn,k) of the enhanced hypercube Qn,k for g=1,2. Furthermore, as the byproduct of our results, we determine the conditional diagnosability of Qn,k under the comparison model and PMC model.

Reliability evaluation of complete cubic networks

Fault diagnostic analysis is extremely important for interconnection networks. Given a graph G and a positive integer g, the g-extra connectivity of G (denoted by ) is the minimum cardinality of a subset S of such that G−S is disconnected and every remaining component has at least g + 1 vertices. The g-extra diagnosability of G (denoted by ), is the maximum number of faulty vertices that the system can guarantee to identify under the condition that every fault-free component contains at least g + 1 vertices. The t/k-diagnosis strategy can detect up to t faulty vertices which might include at most k misdiagnosed vertices. In this paper, we first determine for , , where is an n-dimensional complete cubic network, which generalises the hierarchical cubic network. Moreover, we establish under the PMC model (, ) and under the MM* model (, ), respectively. Furthermore, we show that is -diagnosable under the PMC model. As a consequence, we also derive the related results of the n-dimensional hierarchical cubic network .