Balanced Hypercube Research Articles

The 3-extra conditional diagnosability of balanced hypercubes under MM[formula omitted] model

Diagnosis plays an important role in measuring the reliability of an interconnection network, and the diagnosability of interconnection networks has been widely investigated. For a system G, Zhang et al. introduced h-extra conditional diagnosability, denoted by th̃(G), which is a novel measure of diagnosability. In this paper, several new structural properties of balanced hypercubes have been derived. Based on these properties, the 3-extra conditional diagnosability of an n-dimensional balanced hypercube is determined to be t̃3(BHn)=4n−1 for n≥5 under the MM∗ model.

The reliability analysis based on the generalized connectivity in balanced hypercubes

Recently, network connectivity analysis in terms of reliability has received attention from the network research community. Although traditional connectivity can be used to assess the strength of the connection between two nodes, however, such measures are inadequate for evaluating the connectivity among a set of multiple nodes in a network. Given a set S of vertices in a graph G with |S|⩾2, we say that a tree in G is an S-tree if it connects all vertices of S. Two S-trees T and T′ in G are internally disjoint if E(T)∩E(T′)=0̸ and V(T)∩V(T′)=S. Let κG(S) denote the maximum number of S-trees in G such that every pair of them are internally disjoint. For an integer k⩾2, the generalized k-connectivity of graph G is defined as κk(G)=min{κG(S)∣S⊆V(G)and|S|=k}. In this paper, we investigate the problem of finding the generalized 3-connectivity of the n-dimensional balanced hypercube BHn, which is a hypercube-variant network and is superior to hypercube due to having a smaller diameter. As a result, we prove that κ3(BHn)=2n−1 for n⩾1.

Structure fault tolerance of balanced hypercubes

The S-structure connectivity κ(G;S) and S-substructure connectivity κs(G;S) of graph G with structure graph S, which are newly proposed in recent years, can better measure fault tolerance of networks. In this paper, we study κ(BHn;S) and κs(BHn;S) of n-dimensional balanced hypercube BHn, which is a relatively popular network topology proposed recently, for S∈{Pk,C6α,K1,r} where 4≤k≤n, 6≤6α≤n and 4≤r≤5.

Structure fault tolerance of balanced hypercubes

Let H be a connected subgraph of a given graph G. The H-structure connectivity of G is the cardinality of a minimal set $${\mathcal {F}}$$ of subgraphs of G such that every element in $${\mathcal {F}}$$ is isomorphic to H, and the removal of all the elements of $${\mathcal {F}}$$ will disconnect G. The H-substructure connectivity of graph G is the cardinality of a minimal set $${\mathcal {F}}'$$ of subgraphs of G such that every element in $${\mathcal {F}}'$$ is isomorphic to a connected subgraph of H, and the removal of all the elements of $${\mathcal {F}}'$$ will disconnect G. The two parameters were proposed by Lin et al. in (Theor Comput Sci 634:97–107, 2016), where no restrictions on $${\mathcal {F}}$$ and $${\mathcal {F}}'$$ . In Lu and Wu (Bull Malays Math Sci Soc 43(3):2659–2672, 2020), the authors imposed some restrictions on $${\mathcal {F}}$$ (resp. $${\mathcal {F}}'$$ ) for the n-dimensional balanced hypercube $$\text {BH}_n$$ and requires that two elements in $${\mathcal {F}}$$ (resp. $${\mathcal {F}}'$$ ) cannot share a vertex. Under such restrictions, they determined the (restricted) H-structure and (restricted) H-substructure connectivity of $$\text {BH}_n$$ for $$H\in \{K_1,K_{1,1},K_{1,2},K_{1,3},C_4\}$$ . In this paper, we follow (2016) for the definitions of the two parameters and determine the H-structure and H-substructure connectivity of $$\text {BH}_n$$ for $$H\in \{K_{1,t},P_k,C_4\}$$ , where $$K_{1,t}$$ is the star on $$t+1$$ vertices with $$1\le t\le 2n$$ and $$P_k$$ is a path of length k with $$1\le k\le 7$$ . Some of our main results show that the H-structure connectivity (resp. H-substructure connectivity) of $$\text {BH}_n$$ is equal to the restricted H-structure connectivity (resp. restricted H-substructure connectivity) of $$\text {BH}_n$$ for $$H\in \{K_{1,1},K_{1,2},C_4\}$$ , but the $$K_{1,3}$$ -structure connectivity (resp. $$K_{1,3}$$ -substructure connectivity) of $$\text {BH}_n$$ is not equal to the restricted $$K_{1,3}$$ -structure connectivity (resp. restricted $$K_{1,3}$$ -substructure connectivity) of $$\text {BH}_n$$ unless $$n=\lceil 2n/3\rceil$$ .

Two-disjoint-cycle-cover bipancyclicity of balanced hypercubes

A bipartite graph G is two-disjoint-cycle-cover [r1, r2]-bipancyclic if for any even integer ℓ satisfying r1 ≤ ℓ ≤ r2, there exist two vertex-disjoint cycles C1 and C2 in G such that |V(C1)|=ℓ and |V(C2)|=|V(G)|−ℓ, where |V(G)| denotes the number of vertices in G. In this paper, we study the two-disjoint-cycle-cover bipancyclicity of the n-dimensional balanced hypercube BHn, which is a hypercube-variant network and is superior to hypercube due to having a smaller diameter. As a consequence, we show that BHn is two-disjoint-cycle-cover [4,22n−1]-bipancyclic for n ≥ 2.

Reliability Analysis of Subsystem in Balanced Hypercubes

The probability that a multiprocessor computer system has faults arises as the cardinality of the system grows. The subsystem reliability in a system, defined as the probability that there exists a fault-free subsystem of a specified cardinality when the system has faults. In this paper, we derive an upper bound and a lower bound on the probability of a (n-1)-dimensional subgraph being fault-free in a n-dimensional balanced hypercube under the probabilistic fault model. Numerical simulations indicate that these two analytical results we get are in good consistency, especially when the value of node reliability goes low.

Constructing Two Completely Independent Spanning Trees in Balanced Hypercubes

Constructing Two Completely Independent Spanning Trees in Balanced Hypercubes

On Computing Component (Edge) Connectivities of Balanced Hypercubes

Abstract For an integer $\ell \geqslant 2$, the $\ell $-component connectivity (resp. $\ell $-component edge connectivity) of a graph $G$, denoted by $\kappa _{\ell }(G)$ (resp. $\lambda _{\ell }(G)$), is the minimum number of vertices (resp. edges) whose removal from $G$ results in a disconnected graph with at least $\ell $ components. The two parameters naturally generalize the classical connectivity and edge connectivity of graphs defined in term of the minimum vertex-cut and the minimum edge-cut, respectively. The two kinds of connectivities can help us to measure the robustness of the graph corresponding to a network. In this paper, by exploring algebraic and combinatorial properties of $n$-dimensional balanced hypercubes $BH_n$, we obtain the $\ell $-component (edge) connectivity $\kappa _{\ell }(BH_n)$ ($\lambda _{\ell }(BH_n)$). For $\ell $-component connectivity, we prove that $\kappa _2(BH_n)=\kappa _3(BH_n)=2n$ for $n\geq 2$, $\kappa _4(BH_n)=\kappa _5(BH_n)=4n-2$ for $n\geq 4$, $\kappa _6(BH_n)=\kappa _7(BH_n)=6n-6$ for $n\geq 5$. For $\ell $-component edge connectivity, we prove that $\lambda _3(BH_n)=4n-1$, $\lambda _4(BH_n)=6n-2$ for $n\geq 2$ and $\lambda _5(BH_n)=8n-4$ for $n\geq 3$. Moreover, we also prove $\lambda _\ell (BH_n)\leq 2n(\ell -1)-2\ell +6$ for $4\leq \ell \leq 2n+3$ and the upper bound of $\lambda _\ell (BH_n)$ we obtained is tight for $\ell =4,5$.

Structure and Substructure Connectivity of Balanced Hypercubes

The connectivity of a network directly signifies its reliability and fault-tolerance. Structure and substructure connectivity are two novel generalizations of the connectivity. Let $H$ be a subgraph of a connected graph $G$. The structure connectivity (resp. substructure connectivity) of $G$, denoted by $\kappa(G;H)$ (resp. $\kappa^s(G;H)$), is defined to be the minimum cardinality of a set $F$ of connected subgraphs in $G$, if exists, whose removal disconnects $G$ and each element of $F$ is isomorphic to $H$ (resp. a subgraph of $H$). In this paper, we shall establish both $\kappa(BH_n;H)$ and $\kappa^s(BH_n;H)$ of the balanced hypercube $BH_n$ for $H\in\{K_1,K_{1,1},K_{1,2},K_{1,3},C_4\}$.

The h-edge tolerable diagnosability of balanced hypercubes

To measure the fault diagnosis capability of a multiprocessor system with faulty links, Zhu et al. (2019) [26] introduced the h-edge tolerable diagnosability. This kind of diagnosability is a generalization of the concept of traditional diagnosability. In this paper, as complement to the results in Wei and Xu (2019) [16], we completely determine the h-edge tolerable diagnosability of balanced hypercubes BHn under the PMC model and the MM⁎ model. Thus, the traditional diagnosability of BHn is also determined.

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.

Hamiltonian paths and cycles pass through prescribed edges in the balanced hypercubes

The n-dimensional balanced hypercube BHn (n≥1) has been proved to be a bipartite graph. Let P be a set of edges in BHn. For any two vertices u,v from different partite sets of V(BHn). In this paper, we prove that if |P|≤2n−2 and the subgraph induced by P has neither u nor v as internal vertices , or both of u and v as end-vertices, then BHn contains a Hamiltonian path joining u and v passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. As a corollary, if |P|≤2n−1, then BHn contains a Hamiltonian cycle passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths.

Fault-tolerant-prescribed hamiltonian laceability of balanced hypercubes

Given a bipartite graph G with bipartition (X,Y), let F⊂E(G) be a set of faulty edges and let L be a linear forest in G−F such that |F|+|E(L)|≤k. Let u∈X and v∈Y be any two vertices such that none of the paths in L has u or v as internal vertices or both of them as end vertices. G is said to be k-fault-tolerant-prescribed hamiltonian laceable if G−F admits a hamiltonian path between u and v passing through L. Balanced hypercubes are candidate interconnection networks of multiprocessor systems. In this paper, we prove that the n-dimensional balanced hypercube is (n−1)-fault-tolerant-prescribed hamiltonian laceable.

Paired many-to-many two-disjoint path cover of balanced hypercubes with faulty edges

As a variant of the well-known hypercube, the balanced hypercube $$BH_n$$ was proposed as a desired interconnection network topology for parallel computing. It is known that $$BH_n$$ is bipartite. Assume that $$S=\{s_1,s_2\}$$ and $$T=\{t_1,t_2\}$$ are any two sets of vertices in different partite sets of $$BH_n$$ ( $$n\ge 1$$ ). It has been proved that there exist two vertex-disjoint $$s_1,t_1$$ -path and $$s_2,t_2$$ -path of $$BH_n$$ covering all vertices of $$BH_n$$ . In this paper, we prove that there always exist two vertex-disjoint $$s_1,t_1$$ -path and $$s_2,t_2$$ -path covering all vertices of $$BH_n$$ ( $$n\ge 2$$ ) with at most $$2n-3$$ faulty edges. The upper bound $$2n-3$$ of edge faults can be tolerated is optimal.

Edge-disjoint Hamiltonian cycles of balanced hypercubes

The balanced hypercube, a variant of the hypercube, was proposed as a novel interconnection network of parallel systems. It is known that the balanced hypercube is Hamiltonian. The existence of two edge-disjoint Hamiltonian cycles in balanced hypercubes has remained unknown. In this paper, we prove that the balanced hypercube BHn with n≥2 contains two edge-disjoint Hamiltonian cycles. Further, a linear time algorithm is provided to construct two edge-disjoint Hamiltonian cycles in the balanced hypercube BHn with n≥2.

The (t, k) -Diagnosability of Balanced Hypercube under the PMC Model

ABSTRACTThe network's diagnosability implies its self-diagnostic capability. Compared with classical sequential diagnosis that has been widely used to identify faulty nodes, the -diagnosis is considered as an improvement of sequential diagnosis, where . To be more specific, the -diagnosis can identify and replace at least k faulty nodes in each iteration until all t faulty nodes are found. Meanwhile, compared with the other hypercube graphs, the balanced hypercube (denoted ) has many good properties, such as a better fault-tolerant embedding capability and a smaller diameter. In this paper, we first propose a body discover algorithm (named BD) of , where a body consists of node sets that meet specific conditions. Based on bodies found by BD, we verify that the is -diagnosable. Then we prove that the -diagnosability of is under the PMC model.

Hamiltonian paths passing through prescribed edges in balanced hypercubes

The balanced hypercube was proposed as a novel interconnection network for large-scale parallel systems. It is known that the balanced hypercube is edge-bipancyclic. Given a set of edges P with |P|≤n−1 and two vertices x and y in different bipartite sets, we show that there exists a Hamiltonian path from x to y passing through P in BHn if and only if P is a linear forest and, neither x nor y is an internal vertex of P, and x and y are not end-vertices of a component of P simultaneously. Consequently, the balanced hypercube contains a Hamiltonian cycle passing through a set P of at most n prescribed edges if and only if P is a linear forest. Moreover, our result improves some known results.

Conditional $(t,k)$ -Diagnosis in Regular and Irregular Graphs Under the Comparison Diagnosis Model

Assume that there are at most $t$ faulty vertices. A system is conditionally ( $t,k$ )- diagnosable if at least $k$ faulty vertices (or all faulty vertices if fewer than $k$ faulty vertices remain) can be identified in each iteration under the assumption that every vertex is adjacent to at least one fault-free vertex. Let $\kappa _c(G)$ be the conditional vertex connectivity of $G$ , which measures the vertex connectivity of $G$ according to the assumption that every vertex is adjacent to at least one fault-free vertex. Let $\Delta (G)$ be the maximum degrees of the given graph $G$ . When a graph $G$ satisfies the condition that for any pair of vertices with distance two has at least two common neighbors in $G$ , we show the following two results: 1) An $r$ -regular network $G$ containing $N$ vertices is conditionally $\left(\frac{N+\sqrt{\frac{4\kappa (G)N}{(r+1)(r-1)}}-2}{r+1},\kappa _c(G)\right)$ -diagnosable, where $r \geq 3$ and $N \geq \frac{(r+1)(25r-9)}{4\kappa (G)}$ . 2) An irregular network $G$ containing $N$ vertices is conditionally $(\frac{N}{\Delta (G)+1}-1,\kappa _c(G))$ -diagnosable. By applying the above results to multiprocessor systems, we can measure conditional $(t,k)$ -diagnosabilities for augmented cubes, folded hypercubes, balanced hypercubes, and exchanged hypercubes.

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′}.

On extra connectivity and extra edge-connectivity of balanced hypercubes

Given a graph G and a non-negative integer h, the h-extra connectivity (or h-extra edge-connectivity, resp.) of G, denoted by κh(G) (or λh(G), resp.), is the minimum cardinality of a set of vertices (or edges, resp.) in G, if it exists, whose deletion disconnects G and leaves each remaining component with more than h vertices. In this paper, we obtain a tight upper bound of the h-extra connectivity and the h-extra edge-connectivity of n-dimensional balanced hypercubes BHn for n ≥ 2 and h≤2n−1. As an application, we prove that κ4(BHn)=κ5(BHn)=6n−8 and λ3(BHn)=8n−8, which improves the previously known results given by Yang (2012) and Lü (2017).