Strong Menger connectedness of augmented k-ary n-cubes

A connected graph G is called strongly Menger (edge) connected if for any two distinct vertices x,y of G, there are min{ deg_G(x), deg_G(y)} vertex(edge)-disjoint paths between x and y. In this paper, we consider strong Menger (edge) connectedness of the augmented k-ary n-cube AQ_n,k, which is a variant of k-ary n-cube Q_n^k. By exploring the topological proprieties of AQ_n,k, we show that AQ_n,3 for n≥ 4 (resp. AQ_n,k for n≥ 2 and k≥ 4) is still strongly Menger connected even when there are 4n-9 (resp. 4n-8) faulty vertices and AQ_n,k is still strongly Menger edge connected even when there are 4n-4 faulty edges for n≥ 2 and k≥ 3. Moreover, under the restricted condition that each vertex has at least two fault-free edges, we show that AQ_n,k is still strongly Menger edge connected even when there are 8n-10 faulty edges for n≥ 2 and k≥ 3. These results are all optimal in the sense of the maximum number of tolerated vertex (resp. edge) faults.

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1 Introduction

With continuous advances in technology, a multiprocessor system may contains hundreds or even thousands of processors that communicate by exchanging messages through an interconnection network. The topology of a network can be represented as a graph. Among all fundamental properties for interconnection networks, the connectivity and edge connectivity are the major parameters widely discussed for the connection status of networks.

For a connected graph , the connectivity is the minimum number of vertices removed to get the graph disconnected or trivial; while the edge connectivity is the minimum number of edges removed to get the graph disconnected. Connectivity and edge connectivity are two deterministic measurements for determining the reliability and fault tolerance of a multiprocessor system. In contrast with this concept, Menger [11] provided a local point of view, and defined the connectivity (resp. edge connectivity) of any two vertices as the minimum number of internally vertex-disjoint (resp. edge-disjoint) paths between them.

A connected graph is called strongly Menger (edge) connected if for any two distinct vertices of , there are (edge-)disjoint paths between and . Parallel routing (i.e., construction of disjoint paths or edge-disjoint paths) has been an important issue in the study of computer networks. With the continuous increasing in network size, routing in networks with faults has become unavoidable. Two fault models have been studied for many well-known networks: one is the random fault model, and the other is the conditional fault model (which assumes that the fault distribution is limited). The strong Menger (edge) connectivity of a graph with random faults is defined as follows.

Definition 1.

A graph is called -strongly Menger connected (resp. edge connected) if remains strongly Menger connected (resp. edge connected) for an arbitrary vertex set (resp. edge set ) with .

Note that the term -strong Menger connectivity is also referred to as -fault-tolerant maximal local-connectivity in [1, 2, 15]. Let denote the minimum degree of a graph . The conditional strong Menger (edge) connectivity of a graph is defined as follows.

Definition 2.

A graph is called -conditional strongly Menger connected (resp. edge connected) if remains strongly Menger connected (resp. edge connected) for an arbitrary vertex set (resp. edge set ) with and .

The study on strong Menger vertex/edge connectedness attracts more and more researchers’ attention. On one hand, the (conditional) strong Menger connectivity of many known networks were explored in literature, for example, star graph [12, 9], hypercubes [13], folded hypercubes [21], hypercube-like networks [16], Cayley graphs generated by transposition trees [17], augmented cubes [2], bubble-sort star graph [1], balanced hypercubes [7] etc. On the other hand, the (conditional) strong Menger edge connectivity were investigated on hypercubes [14], folded hypercubes [3], hypercube-like networks [8], balanced hypercubes [7] etc. He et al. [6] and Sabir and Meng [15] presented sufficient conditions for a regular graph to be strongly Menger vertex/edge connected with some restricted conditions. For more information, please refer to [6, 15] and the references therein.

Almost all the known popular classes of networks with strong Menger edge/vertex connectedness are triangle-free. Recently, augmented -ary -cube is proposed for parallel computing by Xiang and Stewart [19]. An augmented -ary -cube is extended from a -ary -cube in a manner analogous to the extension of an -dimensional hypercube to an -dimensional augmented  [4] and has many triangles. Some results about topological properties and routing problems on augmented -ary -cube can be found in [19, 10, 5, 20] etc. In this paper, by exploring and utilizing the structural properties of , we show that (resp. , ) is -strongly (resp. -strongly) Menger connected for (resp. ), and is -strongly Menger edge connected for and . Moreover, under the restricted condition that each vertex has at least two fault-free edges, we show that is -conditional strongly Menger edge connected for and . These results are all optimal with respect to the maximum number of tolerated vertex (resp. edge) faults.

2 Preliminaries

2.1 Notations

Let represent an interconnection network, where a vertex represents a processor and an edge represents a link between vertices and .

Let be the size of vertex set and be the size of edge set. Two vertices and are adjacent if , the vertex is called a neighbor of , and vice versa. For a vertex , let denote a set of vertices in adjacent to , and let . The degree of , denoted by (or ), is the cardinality of . For a vertex set , the neighborhood of in is defined as . When no ambiguity arises, we omit the subscript in the above notations. For any two vertices , we use to denote the number of common neighbors of and in .

A graph is a subgraph of a graph if and . The connected components (simply, component) of a graph are its maximal connected subgraphs. For two disjoint vertex sets or subgraphs and , we use to denote the set of edges with one endpoint in and the other in . For a subset (resp. ), we denote the graph obtained from by removing the vertices (edges) of . In particular, is called a vertex cut (resp. edge cut) of if is disconnected. In this case, the biggest component of is called a large component; the component of which is not the biggest one is called a smaller component.

Given , an -path of length is a finite sequence of distinct vertices such that , and for . A set is an -cut if has no -path. Similarly, a set is an -edge cut if has no -path.

Proposition 1.

([11]) Let be two distinct vertices of a graph .

1. For , the minimum size of an -cut equals the maximum number of disjoint -paths.

2. The minimum size of an -edge cut equals the maximum number of edge-disjoint -paths.

2.2 Augmented k-ary n-cubes

Let and . Assume that all arithmetics on tuple elements are modulo . Xiang and Stewart [19] gave two equivalent definitions of augmented -ary -cube as follows.

Definition 3.

([19]) Let and be integers. The augmented -ary -cube has vertices, each vertex is labelled by an -bit string (or ) with for . There is an edge joining vertex to if and only if one of the following conditions holds.

1. (resp. ) for some and for all ; and the edge is called an -edge (resp. -edge).

2. for some , , ,…, (resp. , ,…, ), for all ; and the edge is called an -edge (resp. -edge).

In the above definition, edges fulfilled the condition (1) and condition (2) are called traditional edges and augmented edges, respectively. Obviously, is a -cycle (i.e., a cycle of length ). Fig. 1 shows , and , where bold lines are traditional edges and dashed lines are augmented edges. In fact, the augmented -ary -cube can also be recursively defined as follows.

Definition 4.

([19]) Fix , augmented -ary -cube has vertex set , and there is an edge joining vertex to if and only if or . Fix . Take copies of augmented -ary -cubes and for th copy add an extra number as the th bit of each vertex (i.e., all vertices have the same th bit if they are in the same copy of augmented -ary -cube). Four more edges are added for each vertex, namely the -edge, -edge, -edge, -edge (as defined in Definition 3).

Lemma 1.

([19]) Let be the augmented -ary -cube, where and are integers.

1. For , the subgraph of induced by vertices with the th bit being , denoted by , is a copy of .

2. is vertex-transitive, is edge-transitive and .

3. Every vertex of has exactly two neighbors in (resp. in ), one is connected by a traditional edge and the other is connected by an augmented edge. Thus .

4. Let be a subset of for . Then .

Two adjacent vertices and are called extra neighbors to each other. Moreover, are called extra edges. From Lemma 1(3), every vertex has exactly four distinct extra neighbors. For convenience, we adopt the following notation to identify the neighbors of a given vertex in . Let . For each , let

,

,

,

.

Remark 1.

([10]) Let and be two distinct vertices in such that and have common neighbors in . Then or . Furthermore, if , then the common neighbors of and in are and if , then the common neighbors of and in are and .

For instance, we consider in (see Fig. 1(c)). Then, we can check that and have common neighbors and . Also, and have common neighbors and .

The following lemma shows the exact number of common neighbors of any two adjacent vertices in .

Lemma 2.

([10]) Let be an edge of . Then the following assertions hold:

1. ;

2. For , ;

3. For ,

4. For and ,

The results are similar with assertions (3) and (4) for and .

The following lemma shows the upper bound of the number of common neighbors for any two distinct vertices in .

Lemma 3.

([10]) Let and be two distinct vertices of . Then the following assertions hold:

1. For , ;

2. For , ;

3. For and , .

Lemma 4.

([5]) Let be the augmented -ary -cubes, where and . If is a subset of with , then .

Wang and Zhao [18] derived the following result which is useful for our proof.

Lemma 5.

([18]) Let be the augmented -ary -cubes, and with . Assume is disconnected. Then

1. has exactly two components, one of which is a singleton or a -cycle, and the vertex set of the -cycle is ;

2. for , has exactly two components, one of which is a singleton.

3 Main Results

In this section, we will consider the (conditional) fault-tolerant strong Menger (edge) connectivity of augmented -ary -cubes. The following result is useful.

Lemma 6.

([15]) Let be an -regular, -connected graph with and . Then is -strongly Menger connected if, for any with , has a component such that .

3.1 Strong Menger connectivity of augmented k-ary n-cubes

In this section, we consider the strong Menger connectivity of augmented -ary -cubes . We will prove that is -strongly Menger connected but not -strongly Menger connected for , and is -strongly Menger connected but not -strongly Menger connected for and .

Lemma 7.

Let be an arbitrary set of vertices in .

1. If for , then has a component such that .

2. If for and , then has a component such that .

Proof. Let be the large component of . First we consider , and . By Lemma 5(2), is connected or has exactly two components, one of which is a singleton. Clearly, if is connected, then and . If is disconnected, then .

Now assume that , and . The proof is by induction on . If , then . Thus, is connected. It leads to and . In what follows, we assume that and the result holds for . Recall that contains disjoint copies of , say , . Let and for . Let and . In addition, we adopt the following notations:

 FI=⋃i∈IFi, FJ=⋃j∈JFj, and AQJn,k=⋃j∈JAQjn,k.

By Lemma 1(2), for . Since , we have . We consider the following cases.

Case 1: .

For all , is connected. By Lemma 1(3), there are edges between subgraphs and . Since for and , there is a fault-free edge between and for each , it implies that is connected. Let . Clearly, .

Case 2: .

Without loss of generality, assume that . By Lemma 1(2), . For , is connected. By Lemma 1(3), there are edges between subgraphs and . For each , since for and , there is a fault-free edge between and . It leads to is connected.

Case 2.1: .

Since , by induction hypothesis on , there exists a component, say in , such that . Since there are edges between subgraphs and and for and , is connected to . Let be the component induced by the vertex set . Then .

Case 2.2: .

In this case, . Let be the large component of and . Obviously, and . By Lemma 1(4), . It leads to , so . If , by Lemma 4, . It leads to

 8n−11≥|F|=f0+|FJ|≥|NAQ0n,kV(M)|+|NAQn,k−AQ0n,k(V(M))|=|NAQn,k(V(M))|≥8n−10,

a contradiction. Thus, and . One can see that is connected to by the similar argument as Case 2.1. Let be the component induced by the vertex set . Then .

Case 3: .

Without loss of generality, assume that , where . By Lemma 1(2), . Since , we have . For , is connected. We consider the following cases.

Case 3.1: or .

Without loss of generality, assume that . Note that and for . For , there is a fault-free edge between and . It leads to is connected. Since every vertex of (resp. ) has two extra neighbors in and , any component of (resp. ) is connected to . Let . Clearly, .

Case 3.2: .

For or , is connected. By the similar argument as Case 3.1, those ’s for and belong to the components, say and , respectively, of . Since every vertex of (resp. ) has four extra neighbors in and , any component of is connected to both (which is part of ) and (which is part of ). This implies that (i.e., and are the same component). By a similar discussion, any component of is contained in both and . This implies that has a large component, say , and . It leads to .

Since is -regular and -connected by Lemma 1(2), combining Lemma 6 and Lemma 7, we have the following result.

Theorem 1.

Let be the augmented -ary -cube. Then

1. is -strongly Menger edge connected for .

2. is -strongly Menger edge connected for and .

Remark 2.

is not -strongly Menger connected for and is not -strongly Menger connected for and . See Fig. 2 for an illustration. Let such that for some and let be a faulty subset of vertices in (i.e., vertices in the darkest area of Fig. 2). By Lemma 2(3), if and , and by Lemma 2(4), if and . We now consider a vertex . Obviously, . Since and some neighbors of are in , the vertices and are not connected with vertex-disjoint paths in . Thus, the results of Theorem 1 are optimal in the sense that the number of faulty vertices cannot be increased.

3.2 Strong Menger edge connectivity of augmented k-ary n-cubes

In this section, we consider the (conditional) strongly Menger edge connectivity of augmented -ary -cubes.

In the following, let be an arbitrary set of edges in . Note that contains disjoint copies of , say , . Let and for . Let and . In addition, we adopt the following notations:

 SI=⋃i∈ISi, SJ=⋃j∈JSj, AQJn,k=⋃j∈JAQjn,k, and sc=|S|−∑i∈[k]0si.

First, we provide two useful lemmas as follows.

Lemma 8.

Let be an arbitrary set of edges in for and . If , then there exists a component in such that .

Proof. Let be the large component of . The proof is by induction on . For , the proof is shown in the Appendix A. In what follows, we assume that and and the result holds for . Recall that and . By Lemma 1(2), for . Since , we have when . The following cases should be considered.

Case 1: .

For all , is connected. By Lemma 1(3), there are edges between subgraphs and . Since for and , there is a fault-free edge between and for each , it implies that is connected. Thus, .

Case 2: .

Without loss of generality, assume that . By Lemma 1(2), . For , is connected. By Lemma 1(3), there are edges between subgraphs and . Since for and , there is a fault-free edge between and for each . It leads to is connected.

Case 2.1: .

Since , by induction hypothesis on , there exists a component, say in , such that