Component Edge Connectivity of Hypercube-like Networks

05/11/2021 ∙ by Dong Liu, et al. ∙ Xiangtan University 0

As a generalization of the traditional connectivity, the g-component edge connectivity cλg(G) of a non-complete graph G is the minimum number of edges to be deleted from the graph G such that the resulting graph has at least g components. Hypercube-like networks (HL-networks for short) are obtained by manipulating some pairs of edges in hypercubes, which contain several famous interconnection networks such as twisted cubes, Mobius cubes, crossed cubes, locally twisted cubes. In this paper, we determine the (g + 1)-component edge connectivity of the n-dimensional HL-networks.

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1 Introduction

As we all know, the interconnection network is an important part of the multiprocessor system. For convenience, we usually model the interconnection network by a graph, with vertices representing processors and edges representing links between processors. There are many parameters to evaluate the reliability of a network. The traditional connectivity of the graph is the most crucial one among them. Generally speaking, the larger the of the network is, the better its reliability is. However, the traditional connectivity only indicates when the network will break but does not further depict the properties of components, which makes it impossible to accurately evaluate the reliability of the network. In order to further describe the properties of components, a lot of more precise connectivity have been proposed, such as extra connectivity [10], super connectivity [2] and restricted connectivity [8]. In 1984, Chartrand et al. [4] and Sampathkumar [14] respectively introduced -component connectivity and -component edge connectivity of the graph .

A -component (edge) cut of a non-complete is a vertex (edge) set to be deleted from the graph such that the resulting graph has at least components. The -component (edge) connectivity of , written (), is the minimum size of the -component (edge) cut of . Obviously, the component (edge) connectivity is a generalization of the traditional connectivity and (). More importantly, compared with the traditional connectivity, the component (edge) connectivity can be better satisfied in practical applications. Thus, many researches have been focused on the component connectivity of some famous networks in recent years. (see, for example [16, 5, 15, 18]). But there are relatively few papers about component edge connectivity. Zhao et al. [17] determined the -component edge connectivity of hypercubes for , . Based on this nice work, we shall be centered on the -component edge connectivity of hypercube-like networks for , .

As a popular topology for the design of the multiprocessor system, hypercubes have many excellent properties including symmetry, relatively diameter, good connectivity and recursive scalability. Improving these properties has led to the evolution of hypercube variants. A lot of hypercube variants, such as twisted cubes [11], crossed cubes [7] and Mbius cubes [6], have been introduced successively. Many of the properties of these networks are identical with that of the hypercubes. In particular, their diameter is shorter than that of the hypercubes. In order to conduct a unified study on these variants, Vaidya et al. proposed hypercube-like networks (in brief, HL-networks), which contain all of the networks mentioned above. Hence, HL-networks attracted considerable attention in the past (see, for example [3, 12]). This class of networks are sometimes called BC-networks [9]. we will use the term HL-networks in this paper.

The recursive definition of the HL-networks is as follows:

and ,
where the symbol represents the perfect matching operation that connects and using some disjoint edges. It’s easy to get that , , , and , where is a cycle of length 4, and and are shown in Figure 1. The -dimensional HL-network is regular, and it has vertices and edges.

Next, we will introduce some notations and definitions which will be used in this paper. Let be a graph. The size of G is the number of edges of . The degree of , denoted by , is the number of edges incident to in . For a subset of , is the subgraph induced by . If and are two disjoint subgraphs of , then we use to denote the set of edges between the subgraphs and . Similarly, we use to denote the set of edges between and , where but . For any , denotes a subgraph obtained by deleting and edges incident to . Similarly, denotes a subgraph obtained by removing all vertices in and all edges incident to vertices of , where is a subgraph of .

The rest of this paper is organized as follows. In section , we shall determine the maximum size of the subgraph induced by vertices in HL-networks. In section , we shall determine the -component edge connectivity of HL-networks by applying the results of section 2.

Figure 1: Two 3-dimensional HL-networks: (a)   (b)

2 The maximum size of the subgraph induced by vertices

It is a classical problem to determine the size (the number of edges) of the subgraph that satisfies some given property in a graph. For instance, Erds has studied an interesting problem in [1]: What is the maximum size of the subgraph without cycles of length 4 in hypercubes. In this section, we shall determine the maximum size of the subgraph induced by vertices in HL-networks. Furthermore, its application in the next section helps us find the -component edge connectivity of HL-networks.

Let be the maximum size of the subgraph induced by vertices in the -dimensional HL-network , that is, . By the mathematical principle of converting decimal digit to binary digit, any integer can be written as the sum of the exponents of 2, that is, , where , for . Li and Yang have determined of hypercubes in [13]. In order to avoid confusion, we use to denote of hypercubes, where . Furthermore, the function has the following property.

Lemma 2.1

[13][17] If , then .

Figure 2: The HL-networks and when s=2

Next, we shall give an algorithm to find the subgraph induced by vertices in the -dimensional HL-network such that .
 
Algorithm-

Input: An -dimensional network , two integer (suppose ) and .

Output: A vertex set and a subgraph .

Initialization: , , where is an empty set.

Iteration: As long as , we take a -dimensional subcube from . Then can be written as , where and are two -dimensional subcubes. Add to , .
 

For ease of understanding, we list as follows (see Figure 2 for ):

We need to verify that the subgraph found by the algorithm- that satisfies and .

Note that and . Thus,

.

By the choice of , one has for . Thus,

.

Theorem 2.2

For any , suppose that with , then .

Proof: For convenience, we define . Firstly, we shall prove for any with by induction on . Clearly, the result holds for since . So we assume . Note that . Let , , for and . Without loss of generality, we suppose . Note . By induction hypothesis, we have that

The third inequality holds because of Lemma 2.1.

Using the algorithm-, we can find a subgraph such that and . Thus, .

Here are some properties of the function .

Lemma 2.3

If , then .

Proof: If , then we can take a subgraph from an ()-dimensional HL-network such that and . Note that is an ()-regular graph. Thus, .

Lemma 2.4

, where .

Proof: If , then . Note that . We have that

.

Otherwise, . Then there is an integer () such that for all . In other words, . Then .

Using the algorithm-, we take two subgraph and from , where and are induced by and vertices respectively. Clearly, and . we can assume that

(1)

where is a -dimensional subcube, is an -dimensional subcube.

In fact, and . In other words, has one less vertex than . Let the vertex be . From (1), we can see that the first subcubes in and are the same. Thus, one has . Clearly, . In addition, there is only one edge between and every in for . Thus, . We have that , that is, .

Lemma 2.5

If , then .

Proof: Suppose , then by Lemma 2.4. Note that . By the Lemma 2.1, .

3 Component edge connectivity of HL-networks

In this section, we shall apply Theorem 2.2 to determine the -component edge connectivity of HL-networks.

For any , we use to denote a set of edges in which each edge has exactly one endpoint in .

Lemma 3.1

For any , let with . Then . Moreover, is strictly increasing (respect to ) for .

Proof: Note that is an -regular graph. We find that . According to Lemma 2.4, if and , then . Hence, we have that , which indicate that is strictly increasing for .

Theorem 3.2

For any , for .

Proof: We first prove that by constructing a -component edge-cut such that . Using the algorithm-, we can get a subgraph such that and . Let , then we have that . Moreover, has at least components. Thus, .

Next, we shall prove that . Assume that is the smallest -component edge-cut and has exactly components. Denote the components in by . Without loss of generality, we can assume that .
Case 1: Suppose that .

In this case, for all . Note that . Then we can find an integer such that but . Let with . Clearly, . It follow that . We suppose that , then . By Lemma 2.3, we have that . Combining with Lemma 3.1, we have that

Note that . Then for . we have that

.

The last inequality holds, because is strictly increasing (respect to ) for by Lemma 3.1.

Case 2: Suppose that .

Let for and let . Set =. If , we can get that by using the proof similar to case 1 . So we assume . If , then the component () is an isolated vertex. Thus, . If , then let . We have that . Since and , we have that

By Lemma 2.5, . Thus, we have that .

Next, we shall show that .

Note that . We take a subgraph of vertices from an ()-dimensional subcube in by using the algorithm-, since . Clearly, . We take a subgraph of vertices from by using the algorithm-, since . So . Use to denote the vertices of . By the proof of Lemma 2.4, there exists a vertex in , say , such that , where . Use to denote vertices of . We assume that (see Figure 3)

Figure 3: The edges between the components

Then we find that

(2)

Let . Note that , and . Thus,

(3)

Moreover, each vertex of has at least two neighbors out of since be taken from an ()-dimensional subcube. Thus, . Note that and . It is not difficult to get that . Thus, by comparing (2) with (3).

To sum up, .

Corollary 3.3

Let be a -component edge-cut of the -dimensional HL-network and , then contains isolated vertices for .

Proof: By the proof of Theorem 3.2, if and only if in case 2, that is, components are isolated vertices.

4 Conclusion

Component edge connectivity is an generation of the traditional connectivity. In this paper, we studied the -component edge connectivity of HL-networks. for . But for , The problem has not been solved.

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