Structure and substructure connectivity of balanced hypercubes

08/06/2018 ∙ by Huazhong Lü, et al. ∙ NetEase, Inc 0

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 κ(G;H) (resp. κ^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 κ(BH_n;H) and κ^s(BH_n;H) of the balanced hypercube BH_n for H∈{K_1,K_1,1,K_1,2,K_1,3,C_4}.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

The interconnection network is crucial in parallel processing and distributed system since the performance of the system is significantly determined by its topology. As the size of a network increases continuously, the reliability and fault-tolerance become central issues. The classical connectivity is an important measure to evaluate fault-tolerance of a network with few processors. An obvious deficiency of the connectivity is the assumption that all the parts of the network can be potentially fail at the same time. However, in large networks, it is unlikely that all the vertices incident to a vertex fail simultaneously, indicating high resilience of large networks. To address the shortcomings of the connectivity stated above, Harary [9] introduced the conditional connectivity of a connected graph by adding some constraints on the components of the resulting graph after vertex deletion. After that, several kinds of conditional connectivity were proposed and investigated [4, 6, 7, 10, 23, 26, 35], such as -connectivity and -connectivity.

The -connectivity of , denoted by , if exists, is defined as the minimum cardinality of a vertex set in , if exists, whose deletion disconnects and leaves each remaining component with at least vertices. The -connectivity of , denoted by ), if exists, is defined as the cardinality of a minimum cardinality of a vertex set in , if exists, whose deletion disconnects and each vertex in the resulting graph has at least neighbors. From the definitions above, it is obvious that and if has and , respectively. So both of -connectivity and -connectivity are generalizations of the connectivity, which supply more accurate measures to evaluate reliability and fault-tolerance of large networks. Moreover, the higher -connectivity or -connectivity the network has, the more reliable the network is [6, 18]. It is known that there exists no polynomial time algorithm to compute the -connectivity and -connectivity of a general graph [2, 6]. The -connectivity [4, 14, 23, 24, 26, 35] and -connectivity [3, 10, 15, 28, 34, 36] of some famous networks are investigated in the literature.

As stated above, most studies on reliability and fault-tolerance of networks are under the assumption that the status of a vertex , whether it is good or faulty, is an event independent of the status of vertices around . In other words, vertices that are linked in a network do not affect each other. Nevertheless, in reality, the neighbors of a faulty vertex might be more vulnerable or have a higher possibility of becoming faulty later. Also note that networks and subnetworks are made into chips. This means that when any vertex is faulty, the whole chip is regarded as faulty. Motivated by these, Lin et al. [16] proposed structure and substructure connectivity to evaluate the fault-tolerance of networks not only from the perspective of individual vertex, but also some special structure of the network.

A set of connected subgraphs of is a subgraph-cut of if is disconnected or trivial. Let be a connected subgraph of , then is an -structure-cut if is a subgraph-cut, and each element in is isomorphic to . The -structure-connectivity of , denoted by , is the minimum cardinality of all -structure-cuts of . Furthermore, is an -substructure-cut if is a subgraph-cut, such that each element in is isomorphic to a connected subgraph of . The -substructure-connectivity of , denoted by , is the minimum cardinality of all -substructure-cuts of .

The balanced hypercube was proposed by Wu and Huang [27]

as a novel interconnection network. As an alternative of the well-known hypercube, the balanced hypercube keeps lots of desirable properties of the hypercube, such as bipartite, high symmetry, scalability, etc. It is known that odd-dimension balanced hypercube has a smaller diameter than that of the hypercube of the same order. In particular, the balanced hypercube is superior to the hypercube in a sense that it supports an efficient reconfiguration without changing the adjacent relationship among tasks

[27]. Some other excellent properties of the balanced hypercube were discussed by many researchers, such as fault-tolerant resource placement problem [11] -connectivity [32, 18, 30] and -connectivity [20], Hamiltonian path (cycle) embedding [5, 8, 29, 31, 13], matching preclusion [17] and matching extendability [19], conditional diagnosability [33] and symmetric properties [37, 38].

Lin et al. [16] considered and of the hypercube for . Later, Sabir and Meng [25] generalized the results in and studied this problem in the folded hypercube. Mane [22] determined with and obtained the upper bound of with . Furthermore, Lv et al. [21] investigated and of the -ary -cube hypercube for . In this paper, we will establish and of the balanced hypercube () for . Note that is a singleton, the -structure connectivity degenerate to traditional connectivity.

The rest of this paper is organized as follows. In Section 2, the definitions of balanced hypercubes and some useful lemmas are presented. The main results of this paper are shown in Section 3. Conclusions are given in Section 4.

2 Preliminaries

Let be a graph, where is vertex-set of and is edge-set of . The number of vertices of is denoted by . The neighborhood of a vertex is the set of vertices adjacent to , written as . Let , we define . For , we use to denote briefly. For other standard graph notations not defined here please refer to [1].

In what follows, we shall give definitions of the balanced hypercube and some lemmas.

Definition 1

.[27] An -dimensional balanced hypercube consists of vertices , where . An arbitrary vertex in has the following neighbors:

  1. mod ,
    mod , and

  2. mod mod ,
    mod mod .

The first coordinate of the vertex in is defined as inner index, and other coordinates outer index.

The following definition shows recursive property of the balanced hypercube.

Definition 2

.[27]

  1. is a -cycle and the vertices are labelled by clockwise.

  2. is constructed from four s, which are labelled by , , , . For any vertex in , its new labelling in is , and it has two new neighbors:

    1. mod mod and

      mod mod if is even.

    2. mod mod and

      mod mod if is odd.

is shown in Fig. 1 (a). Two distinct layouts of are illustrated in Fig. 1 (b) and (c), respectively. Particularly, the layout of in Fig. 1 (c) signifies the ring-like structure of . For brevity, we shall omit “(mod 4)” in the rest of this paper.

Let be a neighbor of in . If and differ only from the inner index, then is called a -dimension edge. If and differ from th outer index (), is called an -dimension edge. It implies from Definition 1 that for each vertex , there exists two -dimension neighbors, , denoted and , where “+” (resp. “”) means that the inner index of (resp. ) is that of plus one (resp. minus one). It can be deduced from Definition 2 that we can divide into four s, , along dimension . It is obvious that the edges between s are -dimension edges. Moreover, each of is isomorphic to . For convenience, we give some symbols as follows.

  • : subset of ;

  • : subset of ;

  • : subset of for each ;

  • : subset of for each ;

  • : subset of .

The following basic properties of the balanced hypercube will be used in the main results of this paper.

Lemma 1

[27]. is bipartite.

By above, vertices of odd (resp. even) inner index are colored with black (resp. white).

Lemma 2

[27]. Vertices and in have the same neighborhood.

Lemma 3

[27, 37]. is vertex-transitive and edge-transitive.

Lemma 4

[18]. Let and be two distinct vertices in . If and have a common neighbor, then and have exact two common neighbors or common neighbors.

Lemma 5

[32]. for .

Lemma 6

[18]. for .

Lemma 7

[30]. for .

Fig. 1: (a) , (b) a layout of and (c) ring-like layout of .

3 Main results

3.1 , , and

It is known that , so we have the following result.

Theorem 8

. and for .

Lemma 9

. and for .

Proof. By vertex-transitivity of , let , and . We set . Clearly, , and for each . It is obvious that . Since , is disconnected and is one of its components. Moreover, each element in is isomorphic to . Thus, the lemma follows.∎

Lemma 10

. If , then is connected.

Proof. We may assume that . Since is 2-connected, is connected if . Thus, we assume that . By Lemma 3, we know that is edge-transitive. So we assume that , and . Let , then is 3-connected. We have the following cases.

Case 1. . It follows that . Obviously, is connected, which implies that is connected.

Case 2. . It follows that . Pick any two adjacent vertices and in , by the ring-like layout of , we can obtain that is 2-connected. After the deletion of any vertex in , the resulting graph is connected. Thus, is connected.

Case 3. . We have . By above, we know that is 2-connected. Let and be any two adjacent vertices in . Moreover, if we delete and from , the resulting graph is also connected. Thus, is connected.

This completes the proof. ∎

Lemma 11

. If for , then is connected.

Proof. We proceed by induction on . We may assume that . By Lemma 10, is connected. Thus, we assume that the statement holds on for each . Next we consider . We set , and and and . Clearly, for each . We consider the following cases.

Case 1. for each . By the induction hypothesis, each is connected. (resp. ) has white (resp. black) vertices, so there are edges between and . Since whenever , there exists a vertex of joining to a vertex of for each . Thus, is connected.

Case 2. for some . We may assume that , therefore, . By the structure of and , there may exist some such that .

Case 2.1. for some . Suppose without loss of generality that . We claim that and . Suppose not. We may assume that . This implies that , a contradiction. Let be the subgraph induced by , then is connected. Note each black (resp. white) vertex in (resp. ) has a neighbor in (resp. ), combining the symmetry of , we only consider white vertices in . Observe that there exists a subset with such that for each , and . Clearly, is a white vertex and is a black vertex. Accordingly, each white vertex of is adjacent to a black vertex in , so it is connected to a vertex in . Thus, is connected.

Case 2.2. for each . Let be the subgraph induced by , then is connected. We shall show that any vertex in is connected to a vertex in via a fault-free path in . We may assume that is a white vertex. Since is triangle-free, and . If or , we are done. Suppose not. Let be the vertex with the same neighborhood of . Thus, we may assume that and for some . Clearly, . Let be the vertex set containing all -dimension neighbors of vertices in . Thus, and the color of vertices in are white. Similarly, for each vertex , and . Additionally, we have . That is, there exists a fault-free path from to a vertex in . Thus, is connected. ∎

Based on Lemmas 9, 10 and 11, we have the following theorem.

Theorem 12

. For , then .

By the definitions of and , we have . So the following statement is straightforward.

Theorem 13

. For , then .

3.2 and

Lemma 14

. and for .

Proof. Let be an arbitrary vertex in . We set . We may assume that if . A -structure-cut of for is illustrated in Fig. 2. Clearly, the subgraph induced by , and is isomorphic to for each . In addition, we have . Since and , is disconnected and is one of components of .X Then the lemma follows.∎

Fig. 2: A -structure-cut for .
Theorem 15

. for .

Proof. We shall show that is connected if . Suppose not. Let be the smallest component of . Note that , then is connected for . It suffices to consider . By Lemma 5, we have whenever . So . Therefore, we assume that . Since is bipartite, for each . Thus, , which implies that there exists a neighbor of in . So we have , a contradiction. Thus, is connected. Combining , we have for . ∎

By Lemma 14 and Theorem15, we have the following result.

Lemma 16

. for .

3.3 and

Lemma 17

. and for .

Proof. Let be an arbitrary vertex in . We set , . A -structure-cut of for is shown in Fig. 3. Clearly, the subgraph induced by , and is isomorphic to , and the subgraph induced by , and is isomorphic to for each . In addition, we have . Since and , is disconnected and is one of components of . Then the lemma follows.∎

Fig. 3: A -structure-cut for .
Theorem 18

. for .

Proof. We shall show that is connected if . Observe that is connected since when . So we assume that . On the contrary, suppose is disconnected if . Let be the smallest component of .

If , then . By Lemma 5, we have . Therefore, we assume that . Since is bipartite, for each . We claim that there exists exact one subgraph of such that . Let the center vertex of be , and pendent vertices be and , respectively. Accordingly, and . Thus, and have three common neighbors, say and . By Lemma 4, and have common neighbors. Therefore, and differ only the inner index. Since there exists exact one vertex such that and differ only the inner index, there exists exact one induced subgraph of such that . Thus, , which implies that there exists a neighbor of in . Thus, is connected.

If , then . By Lemmas 5, 6 and 7, . The proof of is similar to that of . Therefore, we assume that . It follows that contains at least one edge. If , combining is triangle-free, it can be known that . Note that , we have a contradiction. So we assume that . We know that there exists at most two induced subgraphs of such that since each must contain a vertex in that differs only from the inner index of a vertex in . We have whenever . This implies that , a contradiction. Thus, is connected.∎

By Lemma 17 and Theorem18, the following result is straightforward.

Lemma 19

. for .

3.4 and

Lemma 20

. and for .

Proof. Let be an arbitrary vertex in and let be the vertex having the same neighborhood of . We set