Abstract
The connectivity is an important measurement for the fault tolerance of a network. Let be a connected graph with the vertex set and edge set . An -tree of graph is a tree that contains all the vertices in subject to . Two -trees and are internally disjoint if and only if and . Denote by the maximum number of internally disjoint -trees in graph . The generalized -connectivity is a natural generalization of the classical connectivity, which is defined as . In this paper, we mainly focus on the generalized connectivity of cube-connected-cycle and hierarchical hypercube , which were introduced for massively parallel systems. We show that for , and , that is, for any four vertices in (or ), there exist 2 (or ) internally disjoint -trees connecting them in (or ).
1. Introduction
With the emergence and rapid development of high-performance parallel computer technology, people pay more and more attention to the interconnection network of good performance. The excellent topological structure has great advantages in improving reliability. In the design of the network topological structure, fault tolerance is the fundamental consideration, which implies that interconnection network can operate effectively when some nodes and edges fail, that is, it is able to retain some special properties of network. Moreover, the topological structure of an interconnection network can be modeled as an undirected graph , where every vertex of corresponds to a processor, and each edge of corresponds to a communication link. Then, many computer scientists and engineers used some parameters of graph theory to design and analyze topological structures of interconnection networks, such as the connectivity.
The connectivity of a graph is an essential parameter to measure the fault tolerance of an interconnection network. In general, the larger the connectivity of a graph is, the better the fault tolerance is. Let ; if the resulting graph by deleting is disconnected or a trivial graph, is called the cut set. The minimum cardinality of all cut sets of is the connectivity of it, which is denoted as . There is another equivalent definition of the connectivity in [1]. For any vertex set , represents the maximum number of internally disjoint paths joining and in , and . However, the connectivity has one drawback. As the event of vertex failure is random in real multiprocessor networks, it is almost unlikely that all vertices adjacent to the same vertices could fail simultaneously. In order to evaluate the fault tolerance of an interconnection network better, the researchers proposed some generalizations of classical connectivity, for example, the generalized -connectivity, which was introduced in [2].
An -tree of graph is a tree that contains all the vertices of subject to . Two -trees and are internally disjoint if and only if and . Let denote the maximum number of internally disjoint -trees connecting in . For an integer with , the generalized -connectivity of a graph is defined as , which can evaluate the reliability of networks connecting any vertices of . Thus, the generalized 2-connectivity of is exactly the connectivity , namely, .
Many researchers are interested in the generalized -connectivity and obtained some results. For a general graph , Li and Li [3] showed that it is NP-complete to decide whether there are internally disjoint trees connecting , where is a fixed integer and . In addition, the upper and lower bounds of the generalized connectivity were obtained in [4, 5]. For some popular networks, the results on the generalized 3-connectivity of them were given (see [6–11]). However, there are few results on generalized 4-connectivity (see [12, 13]). It is of great significance to study the generalized 4-connectivity. In this paper, we obtain the result of generalized 4-connectivity of cube-connected-cycle and hierarchical hypercube , which are important interconnection networks formed by replacing each vertex of hypercube as a cycle and a hypercube, respectively.
The rest of this paper is organized as follows. In Section 2, we introduce some notations and lemmas used in our proof. In Section 3, we determine the generalized 4-connectivity and generalized 3-connectivity of cube-connected-cycle. In Section 4, we give the generalized 4-connectivity and generalized 3-connectivity. In Section 5, we conclude this paper.
2. Notations and Preliminaries
Let be a simple, undirected graph, and and are denoted as the order and size of , respectively. For a vertex set , is the subgraph of induced by . For a vertex , the set of neighbors of in a graph is denoted by . Let , and we use to denote the subgraph of obtained by removing all the vertices in as well as removing all the edges in from . If , we may write instead of .
Let be the set of binary sequence of length , i.e., . For , the element is called the bitwise complement of , where for each . For an integer , , .
The -dimensional hypercube network is a cube, shortly -cube, with its vertex set , and two vertices are adjacent if and only if they differ exactly in one coordinate. Figure 1 shows . Clearly, is perfect matching of , and . and are two -cubes induced by vertices with the th bit 0 and 1 of , respectively, and then can be denoted as , where the symbol represents the perfect matching operation that connects two different -cubes.
For any and , and are two -cubes induced by the vertices with the th bit 0 and 1 of , respectively. Similarly, and are two -cubes induced by the vertices with the th bit 0 and 1 of , respectively. Hence, , .
By the structure of , the following fact can be obtained easily.
Fact 1. For any two vertices of , say , if , there exists such that or for .
For , the results on the connectivity are obtained in the book [14].
Lemma 1. for .
The generalized 4-connectivity of is shown in [12].
Lemma 2 (see [12]). The generalized 4-connectivity of hypercube is , where .
For an -regular graph, the relation between the generalized -connectivity and -connectivity is investigated in [12].
Lemma 3 (see [12]). Let be an -regular graph. If , then , where .
By Lemmas 2 and 3, we obtain that where .
The upper bound of generalized -connectivity obtained by is given in [5].
Lemma 4 (see [5]). If there are two adjacent vertices of degree , then for .
For any and , -paths denote a set of internally disjoint paths, two endpoints of each path are and , respectively, and the internal vertices belong neither to nor . If , , then -path is a family of internally disjoint paths joining and , which is referred to as a -fan from to .
Lemma 5 (see [14]). Let be a graph with , , , and . Then, there exists a -fan in from to . That is, there exists a family of internally vertex disjoint -paths whose terminal vertices are distinct in .
3. The Generalized 4-Connectivity of
The graph cube-connected-cycle [15] has vertices labeled as , where is an integer between 0 and , called the level of the vertex, and is a binary string of length , called the row of . All arithmetic on indices and levels concerning is assumed to be modulo . Two vertices and are adjacent if and only if either and or and . The latter case means that and differ in exactly the bit in position .
is formed by replacing each vertex of a hypercube graph by a cycle of length . Furthermore, from the definition, one can easily check that all the triangles (resp. quadrangles and pentagons) in (resp. and ) are exactly those cycles replaced on each vertex in in the procedure of creating from for and 5. Figure 2 shows .
Let , be cycles of length of in . Suppose that , and for ; if u is adjacent to u', then, is denoted by the external neighbor of .
Theorem 1. For , .
Proof. Let be any four vertices of . In order to complete the proof, we consider the following cases. Case 1. are distributed in the same cycle. Without loss of generality, we suppose that . As is connected, there is a tree connecting , and in . Denote as the external neighbors of , respectively. Moreover, is connected; then, there is a tree connecting , and , say , . Hence, are the desired trees. Case 2. are distributed in two different cycles. Subcase 2.1. Three vertices of are in one cycle, and the remaining vertex of is in other cycle. Without loss of generality, suppose , . By Lemma 1, , so there are internally disjoint paths joining and , say . Denote , ; without loss of generality, suppose where . Let , , and -paths can be denoted as , where is a path joining and for . As is a cycle and , there is a tree connecting such that has at least 4 vertices. Let and , and the external neighbor of it be , and must be in some for , say . Suppose , is the external neighbor of , and ; clearly, there is a path joining and in , say , so . Denote the external neighbor of by , , , , . As is connected, is also connected, and there is a tree connecting , say , ; then, are the desired trees (see Figure 3). If , we can suppose that , and two desired paths can be found similarly. Subcase 2.2. Two vertices of are in one cycle, and the remaining two vertices of are in other cycle. Without loss of generality, suppose , . The proof is similar to Subcase 2.1. There are two internally disjoint paths joining and in , say , and there is a containing at least three vertices, say . Suppose , and , and the external neighbor of is in , say . As is connected, there is a tree connecting , say ; then, . Let , , , . There is a tree connecting , . Hence, are the desired trees (see Figure 4). Case 3. are distributed in three different cycles. Without loss of generality, we suppose that , , . As for , there are internally disjoint -trees connecting , and in , and each tree contains at least one neighbor of , and . Then, there are internally disjoint trees connecting , and , say , and contains at least one vertex of . Also, ; there exists at most one vertex of not in . There are 2 internally disjoint paths joining and in , say . We can claim that there is such that ; without loss of generality, . Hence, there are two vertices, and , ( can be or ) such that , , where , say . Suppose , , , . There are two paths joining and , say , respectively. Similarly, there are two paths joining and , say , respectively. Let ; then, are the desired trees (see Figure 5). Case 4. are in four different cycles.Without loss of generality, we suppose , , , . By Lemma 2, , and there are internally disjoint trees connecting , and , and each tree contains at least one neighbor of . Then, there are trees connecting , and , say , and contains at least one vertex of . For , there are two vertices, , such that , . Similarly, suppose , , , and , . There are two paths joining and , say and . Similarly, there are two paths joining and , say and , there are two paths joining and , say and , and there are two paths joining and , say and . Then, where , are the desired trees (see Figure 6), and the proof is complete.
From Lemma 3 and Theorem 1, we can obtain the following corollary.
Corollary 1. For , .
4. The Generalized 4-Connectivity of
The hierarchical hypercube was proposed in [16], which is a modification of an -dimension cube-connected-cycle [15], and the cycle is replaced with a hypercube. The definition and some available properties of hierarchical hypercube are introduced as follows.
Definition 1. (see [16]). The vertex set of -dimensional hierarchical hypercube is denoted as . For any vertex of , the adjacency of it is defined as follows: is adjacent to(1) for all .(2), where is the decimal value of .An -dimensional hierarchical hypercube is a -regular bipartite graph of order , where . is triangle-free and consists of clusters, and each cluster is isomorphic to . Let , and be the clusters of . The cross edges between the clusters have the property that every node is incident to exactly one of them (i.e., perfect matching). is shown in Figure 7, and with is shown in Figure 8.
By the definition of hypercube, . For any and , , . Suppose that , and for ; if u is adjacent to u', then, is denoted by the external neighbor of .
In order to obtain the result on , the following lemmas are needed.
As is isomorphic to , for any three vertices of , there must be a path with 4 vertices connecting .
Lemma 6. Let ; there exist internally disjoint trees connecting and each one contains at least 4 vertices.
Proof. It proceeds by induction on . The result holds for . Now we suppose that and the lemma holds for . consists of two different and perfect matching between them, where .
When are distributed in the same , say , as , there are internally disjoint -trees connecting such that each tree has at least 4 vertices by induction hypothesis, say . Let be the external neighbors of in , respectively; there is a tree connecting in , say . Let ; then, are the desired trees.
When are distributed in two , there must be two vertices of , say , such that by Fact 1. Without loss of generality, suppose , and . By Lemma 1, , and there are internally disjoint paths joining and , say . If each contains at least 3 vertices, choose one vertex of , say and is adjacent to . Denote by the external neighbor of . By Lemma 5, there are paths joining and , say , . If there is one path containing only two vertices, that means is adjacent to , say , and contains at least 3 vertices. Clearly, contains at least 3 vertices, and hence contains at least 5 vertices. can be obtained by the proof above.
By Lemma 1, , and there are internally disjoint paths joining and , say , , , and for each .
Lemma 7. For , are internally disjoint paths joining and in , and is denoted as the set consisting of any paths of . Then, is connected.
Proof. We complete the proof by the value of .
First we suppose . By Fact 1, there is such that or for . Without loss of generality, suppose and . Let , , . When , the result can be proved easily. For example, Figure 9 shows the case that and are adjacent. Now we suppose and the lemma holds for , and are shown in Figure 10. If , then there are paths of in , say . As for , must be connected, and is connected. For any vertex not deleted, we can find its external neighbor in , so is connected. If , that means consists of paths of , say . So, for any vertex not deleted, we can find its external neighbor of ; then, is connected.
Next we suppose , and are shown in Figure 11; . , and , . Denote the external neighbor of and in by and , respectively, for . Let be the paths joining and , and , respectively, and be the path joining and , , . Choose any paths of , say . Let . As is connected, and is also connected. Also, , and there is a vertex and its neighbor ; then, is connected. For any vertex , its external neighbor . Furthermore, must be connected. As , there is a vertex and its neighbor is in . Hence, is connected. Also, can be obtained by deleting one path of , is also connected, and the proof is complete.
Lemma 8. For , the set formed by any vertices of and is denoted as and , respectively. There are internally disjoint paths between and .
Proof. It proceeds by induction on . For , without loss of generality, suppose , . As is connected, then there is a path joining and , say , which is denoted as . By Lemma 7, is also connected; then, there is a path joining and . So, are the desired paths joining and .
Now we suppose that and the result holds for . We set , , , and . By induction hypothesis, there are internally disjoint paths joining and , . By Lemma 7, is connected, so there is path joining and , say . The proof is completed.
Theorem 2. Let be two integers with and . Then, .
Proof. Let be any four vertices of . For completing the proof, we consider the following cases. Case 1. are distributed in one cluster. Without loss of generality, we suppose . By Lemma 2, , and there are internally disjoint -trees connecting , and , say . Let be the external neighbors of , respectively, say . As is connected, there is a tree connecting , and , say , . Then, are the desired trees. Case 2. are distributed in two clusters. Subcase 2.1. Three vertices of are in one cluster, and the remaining one vertex of is in another cluster. Without loss of generality, let , . By Lemma 6, there are internally disjoint trees connecting such that each one has at least 4 vertices, say . Choose one vertex of each different from , and , say . For any vertex of , the external neighbor of it must be in some . Let the external neighbor of be ; without loss of generality, for each . Let , and , , . As there are internally disjoint paths joining and in , , there are paths joining and . Clearly, there are internally disjoint paths between and , say . By Lemma 5, there are paths joining and , say , . By Lemma 7, the resulting graph by deleting paths of is connected; then, is connected. Let be the external neighbor of , respectively, and , , , . There is a tree connecting , and in , . are the desired trees (see Figure 12). Subcase 2.2. Two nodes of are in one cluster, and the remaining two vertices of are in another cluster. Without loss of generality, we set , . By Lemma 1, , and there are internally disjoint paths joining , say , and internally disjoint paths joining and , say . Choose two vertices of and , say and , respectively. Let the external neighbor of be , and , and the external neighbor of be , and where . By Lemma 8, there are internally disjoint paths joining and in . Hence, there are paths joining and , say . Then, for . are the desired trees, which are shown in Figure 13. Case 3. are distributed in three clusters. Without loss of generality, we suppose , , . There are internally disjoint paths joining , say . For , there are internally disjoint -trees connecting and , say . We set . Without loss of generality, we suppose for each . So, there must be internally disjoint trees connecting and , say . Choose one vertex of ; then, must be in some . Without loss of generality, suppose , , , and , for each . paths between and are denoted by , and paths between and are denoted by . Then, for (see Figure 14). Case 4. are distributed in four clusters.Without loss of generality, we set , , , and . By , there are internally disjoint -trees connecting , and , say . Let , for . Hence, there are internally disjoint trees connecting , , , and of , say . We set , , for each . The paths joining and are denoted by , the paths joining and are denoted by , the paths joining and are denoted by , and the paths joining and are denoted by . Then, for each (see Figure 15), and the proof is complete.
From Lemma 3 and Theorem 2, the following corollary can be obtained easily.
Corollary 2. Let be two integers with and . Then, .
5. Conclusions
The hypercube is an important interconnection network. is formed by replacing each vertex of a hypercube by a cycle of length . The hierarchical hypercube is a modification of an -dimension cube-connected-cycle, and the cycle is replaced with a hypercube. The generalized -connectivity of interconnection network is a parameter to evaluate the reliability and fault tolerance of it. In this paper, we obtain the generalized 4-connectivity of cube-connected-cycle and hierarchical hypercube. In the future work, the generalized -connectivity of the networks for would be an interesting problem.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the Qinghai Natural Science Foundation of China (no. 2020-ZJ-924) and the National Natural Science Foundation of China (no. 12161073).