The reliability analysis based on the generalized connectivity in balanced hypercubes
Introduction
It is well known that interconnection networks constitute the communication backbone for the multiprocessor system. An interconnection network can be modeled as an undirected connected graph , where represents the set of processors and represents the set of communication links between processors. To evaluate the reliability and fault tolerance of a network in communication, many related concepts and graph parameters have been put forward. As usual, one can use traditional connectivity to assess the strength of the connection between two nodes in a network. Past and recent analyses revealed that the classic connectivity metrics are insufficient to determine how well-connected or well-designed a network is against the node or link failures [14], [25]. In particular, it is important to evaluate the connectivity among a set of multiple nodes in a distributed network when we are in an application for multi-party computation or communication [1], [3]. Hence, the generalized connectivity as one generalization of the traditional connectivity via some tree structures connecting more than two nodes is of great significance.
The connectivity of a connected graph , denoted by , is the minimum number of vertices whose deletion results in a disconnected graph. A graph is -connected if . A well-known result by Whitney [38] (as a corollary of Menger’s Theorem [34]) provided an equivalent definition of connectivity as follows: A graph is -connected if and only if there exist internally disjoint paths between any two vertices in . Given a set with , a tree in is said to be an -tree if it connects all vertices of . For an integer , the generalized -connectivity of a graph is defined as , where denotes the maximum number of -trees in such that every pair of them, say and , are internally disjoint (i.e., and ). Clearly, the generalized -connectivity is exactly the traditional connectivity, i.e., .
The generalized -connectivity was firstly introduced by Hager [11], which has been received much attention in recent years. Li et al. [22] proved that is not always true in general case, while for every connected graph . Currently, the exact values of are known only for small classes of graphs, such as complete graphs [4], complete bipartite graphs [17], [35], complete equipartition 3-partite graphs [18]. In addition, it has pointed out in [16], [22] that the investigation of for general is very difficult (see also [5] for complexity results). Thus, many pieces of research focused on the study of or . For example, results of were obtained for product graphs [21], [23], [24], star graphs and bubble-sort graphs [27], alternating group graphs and -star graphs [47], -bubble-sort graphs [52], recursive circulants [28], data center networks DCell [12], line graph and the total graph of the complete bipartite graph [15], Cayley graphs generated by trees and cycles [26], Cayley graphs generated by complete graphs and wheel graphs [49], and -regular -connected graphs with recursive structure [51]. Also, results of are known for hypercubes [31], exchanged hypercubes [48], and dual cubes [50]. For more works related to , one may also refer to [19], [20], [36], [37] and a monograph [25].
The -dimensional balanced hypercube , proposed by Wu and Huang [39], is a hypercube-variant network with high symmetry and superior properties. is a bipartite graph with vertex-symmetry [39] and edge-symmetry [53]. So far, there are many research results of balanced hypercubes. For example, the explorations of (conditional) matching preclusion number [32], completely independent spanning trees [44], diverse connectivities and edge connectivities [9], [30], [43], [53], diagnosabilities in distinct models [10], [41], [42], [46]. In particular, research on path and cycle (fault-tolerant) embeddings have many applications [6], [7], [8], [13], [29], [33], [40], [45]. To the best of our knowledge, there is no research result to deal with the generalized connectivity of .
In this paper, we will consider the generalized 3-connectivity of . Due to the recursive structure, consists of four copies of , denoted by for (see Definition 2 in the next section). However, the relationships between pairs of copies of are distinct. For example, there exist edges between and , while there are no edges between and , which makes the discussion in constructing -trees more difficult. Moreover, since is a bipartite graph, each copy contains two kinds of vertices (called white vertices and black vertices, respectively) whose neighbors outside the copy are completely different. Thus, the situations in the proof of become more complex. As a consequence, we determine the following result.
Theorem 1 for .
Section snippets
Preliminaries
Let be a simple undirected graph. The subgraph of induced by a subset is denoted by , and the subgraph is denoted by . Let stand for the number of edges incident with a vertex in . A graph is -regular if for all . We denote by the minimum degree of the vertices of . For any two vertices , an -path is a path starting at and ending at . Two -paths and are internally disjoint if and
Proof of Theorem 1
As is -regular, by Lemma 8, we have . We now prove that by induction on . For , the proof is trivial. For , by Lemma 4, . By Lemma 9, , it implies that the result holds for . Assume that and the result of Theorem 1 holds for .
Let be an arbitrary 3-subset of . The sets of the out-neighbors of , and are denoted by , and , respectively. By Lemma 3(1), without loss of generality,
Concluding remarks
Connectivity is one of the most important concepts in graph theory, and it has been extensively used for measuring the reliability of networks in theoretical computer science. To meet the need for a more refined measure of the reliability in networks, the concept of generalized connectivity is proposed. In this paper, we determine the generalized 3-connectivity of the -dimensional balanced hypercube . Even though has the recursive structure with four copies of , the relationship
Acknowledgments
This work was supported by the National Natural Science Foundation of China under the grants 11971054 and 11731002 (R.-X. Hao) and by the Ministry of Science and Technology of Taiwan under the grant 107-2221-E-141-001-MY3 (J.-M. Chang). The authors express their sincere thanks to the anonymous referees and editors for their valuable suggestions.
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2021, Discrete Applied MathematicsCitation Excerpt :It also supports an efficient reconfiguration without changing the adjacent relationship among tasks [25]. The other properties of balanced hypercubes attract many researchers’ study recently years, such as cycle embedding properties [3–5,21,34], extra connectivity [30], super connectivity [28], symmetric property [39], fault-tolerant hamiltonian laceability [38], reliability analysis [22], structure fault tolerance [15], and edge tolerable diagnosability [27]. [25]