Fault-free Edges Research Articles

Reliability assessment for k-ary n-cubes with faulty edges

The g-restricted edge connectivity is an important measurement to assess the reliability of networks. The g-restricted edge connectivity of a connected graph G is the minimum size of a set of edges in G, if it exists, whose deletion separates G and leaves every vertex in the remaining components with at least g neighbors. The k-ary n-cube is an extension of the hypercube network and has many desirable properties. It has been used to build the architecture of the Supercomputer Fugaku. This paper establishes that for g≤n, the g-restricted edge connectivity of 3-ary n-cubes is 3⌊g/2⌋(1+(gmod2))(2n−g), and the g-restricted edge connectivity of k-ary n-cubes with k≥4 is 2g(2n−g). These results imply that in Qn3 with at most 3⌊g/2⌋(1+(gmod2))(2n−g)−1 faulty edges, or Qnk(k≥4) with at most 2g(2n−g)−1 faulty edges, if each vertex is incident with at least g fault-free edges, then the remaining network is connected.

Cycle Embedding in Enhanced Hypercubes with Faulty Vertices

The enhanced hypercube is a well-known variant of the hypercube and can be constructed from a hypercube by adding an edge to every pair of vertices with complementary addresses. Let Fv denote the set of faulty vertices in an n-dimensional enhanced hypercube Qn,k(1≤k≤n−1). In this paper, we conclude that if n≥2, then every fault-free edge of Qn,k−Fv lies on a fault-free cycle of every even length from 4 to 2n−2|Fv|, and if n(≥2) and k have the different parity, then every fault-free edge of Qn,k−Fv lies on a fault-free cycle of every possible odd length from n−k+4 to 2n−2|Fv|−1, where |Fv|≤n−2.

Cycle Existence for All Edges in Folded Hypercubes under Scope Faults

The reliability of large-scale networks can be compromised by various factors such as natural disasters, human-induced incidents such as hacker attacks, bomb attacks, or even meteorite impacts, which can lead to failures in scope of processors or links. Therefore, ensuring fault tolerance in the interconnection network is vital to maintaining system reliability. The n-dimensional folded hypercube network structure, denoted as FQn, is constructed by adding an edge between every pair of vertices with complementary addresses from an n-dimensional hypercube, Qn. Notably, FQn exhibits distinct characteristics based on the dimensionality: it is bipartite for odd integers n≥3 and non-bipartite for even integers n≥2. Recently, in terms of the issue of how FQn performs in communication under regional or widespread destruction, we mentioned that in FQn, even when a pair of adjacent vertices encounter errors, any fault-free edge can still be embedded in cycles of various lengths. Additionally, even when the smallest communication ring experiences errors, it is still possible to embed cycles of any length. The smallest communication ring in FQn is observed to be the four-cycle ring. In order to further investigate the communication capabilities of FQn, we further discuss whether every fault-free edge will still be a part of every communication ring with different lengths when the smallest communication ring is compromised in FQn. In this study, we consider a fault-free edge e=(u,v) and F4={f1, f2,f3, f4} as the set of faulty extreme vertices for any four cycles in FQn. Our research focuses on investigating the cycle-embedding properties in FQn−F4, where the fault-free edges play a significant role. The following properties are demonstrated: (1) For n≥4 in FQn−F4, every even length cycle with a length ranging from 4 to 2n−4 contains a fault-free edge e; (2) For every even n≥4 in FQn−F4, every odd length cycle with a length ranging from n+1 to 2n−5 contains a fault-free edge e. These findings provide insights into the cycle-embedding capabilities of FQn, specifically in the context of fault tolerance when considering certain sets of faulty vertices.

Local diagnosability of bipartite graphs with conditional faulty edges under Preparata, Metze and Chien’s model

Diagnosability of a multiprocessor system is an important research topic. The system or interconnection network has an underlying topology, which usually presented by a graph. In this paper, let G be a bipartite graph with δ ( G ) = δ and let there be at most two common neighbor vertices of any two vertices in G under the PMC model. We firstly study the diagnosability of G . We prove that G − F keeps the strong local diagnosability property even if it has the set F of ( δ − 2 ) faulty edges. Secondly, we study the diagnosability of G with conditional faulty edges. We prove that G − F keeps strong local diagnosability property even if it has the set F of ( 3 δ − 7 ) faulty edges, provided that each vertex of G − F is incident with at least two fault-free edges. Finally, we prove that G − F keeps strong local diagnosability property no matter how many edges are faulty, provided that each vertex of G − F is incident with at least three fault-free edges.

The Local Diagnosability of a Class of Cayley Graphs with Conditional Faulty Edges Under the PMC Model

Abstract The diagnosability of a multiprocessor system is of great significance in measuring the reliability and faulty tolerance of interconnection networks. In this paper, we firstly study the diagnosability of a class of Cayley graphs $Cay(H_n,S_n)$ under the PMC model. We prove that $Cay(H_n,S_n)-F$ keeps the strong local diagnosability property even if it has the set $F$ of $(m-2)$ faulty edges and $m-2$ is maximum number of faulty edges, where $m$ is the regular degree of $Cay(H_n,S_n)$. Secondly, we study the diagnosability of $Cay(H_n,S_n)$ with conditional faulty edges under the PMC model. We prove that $Cay(H_n,S_n)-F$ keeps strong local diagnosability property even if it has the set $F$ of $(3m-10)$ faulty edges, provided that each vertex of $Cay(H_n,S_n)-F$ is incident with at least two fault-free edges, where $3m-10$ is maximum number of faulty edges. Finally, we prove that $Cay(H_n,S_n)-F$ keeps strong local diagnosability property no matter how many edges are faulty, provided that each vertex of $Cay(H_n,S_n)-F$ is incident with at least four fault-free edges.

The Local Diagnosability of Folded Hypercubes with Conditional Faulty Edges Under the PMC Model

The diagnosability of a multiprocessor system is of great significance in measuring the reliability and faulty tolerance of interconnection networks. In this paper, we firstly study the diagnosability of the [Formula: see text]-dimensional folded hypercube [Formula: see text] under the PMC model. We prove that [Formula: see text] keeps the strong local diagnosability property even if it has the set [Formula: see text] of [Formula: see text] faulty edges and [Formula: see text] is maximum number of faulty edges. Secondly, we study the diagnosability of [Formula: see text] [Formula: see text] and [Formula: see text] is odd) with conditional faulty edges under the PMC model. We prove that [Formula: see text] keeps strong local diagnosability property even if it has the set [Formula: see text] of [Formula: see text] faulty edges, provided that each vertex of [Formula: see text] is incident with at least two fault-free edges, where [Formula: see text] is maximum number of faulty edges. Finally, we prove that [Formula: see text] keeps strong local diagnosability property no matter how many edges are faulty, provided that each vertex of [Formula: see text] is incident with at least three fault-free edges.

The 3-Good Neighbor Edge-Fault-Tolerant Strong Menger Edge Connectivity on the Class of BC Networks

A graph [Formula: see text] is called strongly Menger edge connected (SM-[Formula: see text] for short) if the number of disjoint paths between any two of its vertices equals the minimum degree of these two vertices. In this paper, we focus on the maximally edge-fault-tolerant of the class of BC-networks (contain hypercubes, twisted cubes, Möbius cubes, crossed cubes, etc.) concerning the SM-[Formula: see text] property. Under the restricted condition that each vertex is incident with at least three fault-free edges, we show that even if there are [Formula: see text] faulty edges, all BC-networks still have SM-[Formula: see text] property and the bound [Formula: see text] is sharp.

Every edge lies on cycles of folded hypercubes with a pair of faulty adjacent vertices

An n-dimensional folded hypercube FQn is a well-known extended structure of an n-dimensional hypercube Qn, which can be constructed from Qn by adding an edge to every pair of vertices with complementary addresses. FQn possesses many properties that are superior to those of Qn including the diameter, fault diameter, and connectivity. It has been shown that FQn is bipartite for every odd n≥3, and non-bipartite for every even n≥2. In this report, we let {f1,f2} denote the set of a pair of adjacent faulty vertices in FQn. We then show the cycle embedding properties as follows:1. For n≥3, every fault-free edge can lie on a cycle of any even length l with 4≤l≤2n−4 in FQn−{f1,f2};2. For every even n≥4, every fault-free edge can lie on a cycle of every odd length l with n+1≤l≤2n−3 in FQn−{f1,f2}.Our study improves the result of the previous study with respect to the embedding of the longest cycle in a non-bipartite FQn, and the quantity of the faulty vertices in FQ3.

(n − 2)-Fault-Tolerant Edge-Pancyclicity of Crossed Cubes CQn

As one of the most fundamental networks for parallel and distributed computation, cycle is suitable for developing simple algorithms with low communication cost. A graph [Formula: see text] is called [Formula: see text]-fault-tolerant edge-pancyclic if after deleting any faulty set [Formula: see text] of [Formula: see text] vertices and/or edges from [Formula: see text], every correct edge in the resulting graph lies in a cycle of every length from [Formula: see text] to [Formula: see text], inclusively, where [Formula: see text] is the girth of [Formula: see text], the length of a shortest cycle in [Formula: see text]. The [Formula: see text]-dimensional crossed cube [Formula: see text] is an important variant of the hypercube [Formula: see text], which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant edge-pancyclicity of [Formula: see text], and shows that if [Formula: see text] contains at most [Formula: see text] faulty vertices and/or edges then, for any fault-free edge [Formula: see text] and every length [Formula: see text] from [Formula: see text] to [Formula: see text] except [Formula: see text], there is a fault-free cycle of length [Formula: see text] containing the edge [Formula: see text]. The result is optimal in some senses.

Hamiltonian Cycle Embeddings in Faulty Hypercubes Under the Forbidden Faulty Set Model

In this paper, we study the fault-tolerant capability of hypercubes with respect to the hamiltonian property based on the concept of forbidden faulty sets. We show, with the assumption that each vertex is incident with at least three fault-free edges, that an [Formula: see text]-dimensional hypercube contains a fault-free hamiltonian cycle, even if there are up to [Formula: see text] edge faults. Moreover, we give an example to show that the result is optimal with respect to the number of edge faults tolerated.

Fault-free Hamiltonian cycle including given edges in folded hypercubes with faulty edges

The folded hypercube is an important interconnection network for multiprocessor systems. Let [Formula: see text] with [Formula: see text] denote an [Formula: see text]-dimensional folded hypercube. For a given fault-free edge set [Formula: see text] with [Formula: see text] and a faulty edge set [Formula: see text] with [Formula: see text], in this paper we prove that [Formula: see text] contains a fault-free Hamiltonian cycle including each edge of [Formula: see text] if and only if the subgraph induced by [Formula: see text] is linear forest. Furthermore, we give the definitions of the distance among three vertex-disjoint edges and the distance between a vertex and a vertex set. For three vertex-disjoint edges [Formula: see text], the distance among them is denoted by [Formula: see text]. For a vertex [Formula: see text] and a vertex set [Formula: see text], the distance between [Formula: see text] and [Formula: see text] is denoted by [Formula: see text].

Fault-Free Hamiltonian Cycles in Balanced Hypercubes with Conditional Edge Faults

The balanced hypercube, [Formula: see text], is a variant of hypercube [Formula: see text]. Zhou et al. [Inform. Sci. 300 (2015) 20–27] proposed an interesting problem that whether there is a fault-free Hamiltonian cycle in [Formula: see text] with each vertex incident to at least two fault-free edges. In this paper, we consider this problem and show that each fault-free edge lies on a fault-free Hamiltonian cycle in [Formula: see text] after no more than [Formula: see text] faulty edges occur if each vertex is incident with at least two fault-free edges for all [Formula: see text]. Our result is optimal with respect to the maximum number of tolerated edge faults.

Intermittent Fault Diagnosability of Hyper Petersen Network

The problem of permanent fault diagnosis has been discussed widely, and the diagnosability of many well-known networks have been explored. Faults of a multiprocessor system generally include permanent and intermittent, with intermittent faults regarded as the most challenging to diagnose. In this paper, we investigate the intermittent fault diagnosability of hyper Petersen networks. First, we derive that an $$n$$ -dimensional hyper Petersen network $$HP_{n}$$ with fault-free edges is $$(n - 1)_{i}$$ -diagnosable under the PMC model. Then, we investigate the intermittent fault diagnosability of $$HP_{n}$$ with faulty edges under the PMC model. Finally, we prove that an $$n$$ -dimensional hyper Petersen network $$HP_{n}$$ is $$(n - 2)_{i}$$ -diagnosable under the MM* model.

Cycles embedding in balanced hypercubes with faulty edges and vertices

Wu and Huang proposed a new variation of the hypercube, named balanced hypercube, which possesses many good properties such as bipanconnectivity, edge-bipancyclicity, Hamiltonian laceability, hyper Hamiltonian laceability. In this paper, we consider n-dimensional balanced hypercube with |Fe| faulty edges and |Fv| faulty vertices. We prove that if |Fv|+|Fe|≤n−1, then every fault-free edge of BHn lies on a fault-free cycle of every even length from 6 to 22n−2|Fv|, where n≥2; and if |Fv|+|Fe|≤2n−3, then there is a fault-free cycle of every even length from 6 to 22n−2|Fv| in BHn, where n≥2. Furthermore, we propose the distance between vertex-disjoint edge e and cycle C, i.e., d(e,C)=min{d(e,e′)|e′∈E(C)}, where d(e,e′)=min{d(u,x),d(u,y),d(v,x),d(v,y)|(u,v)=e,(x,y)=e′}.

Fault-tolerant strong Menger (edge) connectivity and 3-extra edge-connectivity of balanced hypercubes

A connected graph G is called strongly Menger (edge) connected if for any two distinct vertices x,y of G, there are min⁡{degG⁡(x),degG⁡(y)}(-edge)-disjoint paths between x and y. The g(g≥0)-extra edge-connectivity of the connected graph G, denoted by λg(G), is the minimal cardinality of a set of edges of G, if exists, whose deletion disconnects G and each remaining component contains more than g vertices. In this paper, we show that the n-dimensional balanced hypercube BHn, which is a variant of hypercube Qn, is still strongly Menger (edge) connected even when there are (2n−4) faulty vertices (resp. (2n−2) faulty edges) for n≥2. Moreover, under the restricted condition that each vertex has at least two fault-free edges, we show that BHn is still strongly Menger edge connected even when there are (6n−8) faulty edges for n≥2. These results are all optimal with respect to the maximum number of tolerated (edge) faults. Furthermore, we showed that the 3-extra edge-connectivity of BHn is 8n−8 for n≥2.

Cycles embedding in folded hypercubes under the conditional fault model

A faulty network G is under the conditional fault model, i.e., every fault-free vertex of G is incident to at least two fault-free edges. Let FFv and FFe be the set of faulty vertices and faulty edges in FQn, respectively. In this paper, we consider FQn under the conditional fault model and prove that if |FFv|+|FFe|≤2n−4 and n≥3, then FQn−FFv−FFe contains a fault-free cycle of every even length from 4 to 2n−2|FFv|; if |FFv|+|FFe|≤2n−5 and n≥4 is even, then FQn−FFv−FFe contains a fault-free cycle of every odd length from n+1 to 2n−2|FFv|−1.

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.

Edge-pancyclicity and edge-bipancyclicity of faulty folded hypercubes

Let Fv and Fe be sets of faulty vertices and faulty edges, respectively, in the folded hypercube FQn so that |Fv|+|Fe|≤n−2, for n≥2. Choose any fault-free edge e. If n≥3 then there is a fault-free cycle of length l in FQn containing e, for every even l ranging from 4 to 2n−2|Fv|; if n≥2 is even then there is a fault-free cycle of length l in FQn containing e, for every odd l ranging from n+1 to 2n−2|Fv|−1.

Every edge lies on cycles embedding in folded hypercubes with both vertex and edge faults

The folded hypercube is a well-known variation of hypercube structure and can be constructed from a hypercube by adding an edge to every pair of vertices with complementary addresses. Let [Formula: see text] (respectively, [Formula: see text]) denote the set of faulty vertices (respectively, faulty edges) in an [Formula: see text]-dimensional folded hypercube [Formula: see text]. In the case that all edges in [Formula: see text] are fault-free, Cheng et al. [Cycles embedding on folded hypercubes with faulty vertices, Discrete Appl. Math. 161 (2013) 2894–2900] has shown that (1) every fault-free edge of [Formula: see text] lies on a fault-free cycle of every even length from [Formula: see text] to [Formula: see text] if [Formula: see text], where [Formula: see text]; and (2) every fault-free edge of [Formula: see text] lies on a fault-free cycle of every odd length from [Formula: see text] to [Formula: see text] if [Formula: see text], where [Formula: see text] is even. In this paper, we extend Cheng’s result to obtain two further properties, which consider both vertex and edge faults, as follows: (1) Every fault-free edge of [Formula: see text] lies on a fault-free cycle of every even length from [Formula: see text] to [Formula: see text] if [Formula: see text], where [Formula: see text]; (2) Every fault-free edge of [Formula: see text] lies on a fault-free cycle of every odd length from [Formula: see text] to [Formula: see text] if [Formula: see text], where [Formula: see text] is even.