The Component Connectivity of Alternating Group Graphs and Split-Stars

12/03/2018 ∙ by Mei-Mei Gu, et al. ∙ BEIJING JIAOTONG UNIVERSITY 0

For an integer ℓ≥ 2, the ℓ-component connectivity of a graph G, denoted by κ_ℓ(G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least ℓ components or a graph with fewer than ℓ vertices. This is a natural generalization of the classical connectivity of graphs defined in term of the minimum vertex-cut and is a good measure of robustness for the graph corresponding to a network. So far, the exact values of ℓ-connectivity are known only for a few classes of networks and small ℓ's. It has been pointed out in [Component connectivity of the hypercubes, Int. J. Comput. Math. 89 (2012) 137--145] that determining ℓ-connectivity is still unsolved for most interconnection networks, such as alternating group graphs and star graphs. In this paper, by exploring the combinatorial properties and fault-tolerance of the alternating group graphs AG_n and a variation of the star graphs called split-stars S_n^2, we study their ℓ-component connectivities. We obtain the following results: (i) κ_3(AG_n)=4n-10 and κ_4(AG_n)=6n-16 for n≥ 4, and κ_5(AG_n)=8n-24 for n≥ 5; (ii) κ_3(S_n^2)=4n-8, κ_4(S_n^2)=6n-14, and κ_5(S_n^2)=8n-20 for n≥ 4.

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1 Introduction

An interconnection network is usually modeled as a connected graph , where the vertex set represents the set of processors and the edge set represents the set of communication channels between processors. For a subset , the graph obtained from by removing all vertices of is denoted by . In particular, is called a vertex-cut of if is disconnected. The connectivity of a graph , denoted by , is the cardinality of a minimum vertex-cut of , or is defined to be when is a complete graph. For making a more thorough study on the connectivity of a graph to assess the vulnerability of its corresponding network, a concept of generalization was first introduced by Chartrand et al. [7]. For an integer , the generalized -connectivity of a graph , denoted by , is the minimum number of vertices whose removal from results in a graph with at least components or a graph with fewer than vertices. For such a generalization, a synonym was also called the general connectivity [33] or -component connectivity [27]. Since there exist diverse definitions of generalized connectivity in the literature (e.g., see [23, 24]), hereafter we follow the use of the terminology “-component connectivity” (or -connectivity for short) to avoid confusion.

The -connectivity is concerned with the relevance of the cardinality of a minimum vertex-cut and the number of components caused by the vertex-cut. Accordingly, finding -connectivity for certain interconnection networks is a good measure of robustness for such networks. So far, the exact values of -connectivity are known only for a few classes of networks and small ’s. For example, -connectivity is determined on hypercube for (see [27]) and (see [42]), folded hypercube for (see [41]), dual cube for (see [40]), hierarchical cubic network for (see [17]), complete cubic network for (see [18]), and generalized exchanged hypercube for and (see [19]). Note that the number of vertices of graphs in the above classes is an exponent related to . Also, it has been pointed out in [27] that determining -connectivity is still unsolved for most interconnection networks such as star graphs and alternating group graphs . The closest results for the two classes of graph were given in [15, 16], but these are asymptotic results. Recently, Chang et al. [3, 4] determined the -connectivity of alternating group networks for . Note that the two classes of and are definitely different.

In this paper, we study -connectivity of the -dimensional alternating group graph and the -dimensional split-stars (defined later in Section 2), which were introduced by Jwo et al. [28] and Cheng et al. [14], respectively, for serving as interconnection network topologies of computing systems. The two families of graphs have received much attention because they have many nice properties such as vertex-transitive, strongly hierarchical, maximally connected (i.e., the connectivity is equal to its regularity), and with a small diameter and average distance. In particular, Cheng et al. [12] showed that alternating group graphs and split-stars are superior to the -cubes and star graphs under the comparison using an advanced vulnerability measure called toughness, which was defined in [20]. For the two families of graphs, many researchers were attracted to study fault tolerant routing [10], fault tolerant embedding [5, 6, 37], matching preclusion [2, 9], restricted connectivity [13, 22, 31, 30, 39] and diagnosability [8, 22, 25, 29, 30, 31, 36]. Moreover, alternating group graphs are also edge-transitive and possess stronger and rich properties on Hamiltonicity (e.g., it has been shown to be not only pancyclic and Hamiltonian-connected [28] but also panconnected [6], panpositionable [35] and mutually independent Hamiltonian [34]). The following structural property disclosed by Cheng et al. [16] is of particular interest and closely related to -component connectivity. They showed that even though linearly many faulty vertices are removed in , the rest of the graph has still a large connected component that contains almost all the surviving vertices. Therefore, this component can be used to perform original network operations without degrading most of its capability. For more further investigations on alternating group graphs and split-stars, see also [11, 38, 44].

In this paper, we determine -component connectivity for of the -dimensional alternating group graph and -dimensional split-star as follows.

Theorem 1.

and for , and for .

Theorem 2.

, , and for .

2 Preliminaries

For , let and be a permutation of elements of , where is the symbol at the position in the permutation. Two symbols and are said to be a pair of inversion of if and . A permutation is an even permutation provided it has an even number of inversions. Let (resp., ) denote the set of all permutations (resp., even permutations) over . An operation acting on a permutation that swaps symbols at positions and and leaves all other symbols undisturbed is denoted by . The composition means that the operation is taken by swapping symbols at positions and , and then swapping symbols at positions and . For , we further define two operations, and on by setting and . Accordingly, (resp., ) is the permutation obtained from by rotating symbols at positions and from left to right (resp., from right to left). Taking as an example, if , then and .

Recall that the Cayley graph on a finite group with respect to a generating set of is defined to have the vertex set and the edge set . We now formally give the definition of alternating group graphs and split-stars as follows.

Definition 1.

(see [28]) The -dimensional alternating group graph, denoted by , is a graph consisting of the vertex set and two vertices are adjacent if and only if for some . That is, with .

A path (resp., cycle) of length is called a -path (resp., -cycle). Clearly, from the above definition, is isomorphic to a 3-cycle. As a Cayley graph, is vertex-transitive. Also, it has been shown in [28] that contains vertices, edges, and is an edge-transitive and -regular graph with diameter . It is well known that every edge-transitive graph is maximally connected, and hence . For and , let be the subgraph of induced by vertices with the rightmost symbol . Like most interconnection networks, can be defined recursively by a hierarchical structure. Thus, is composed of disjoint copies of for , and each is isomorphic to . If a vertex belongs to a subgraph , we simply write instead of . An edge joining vertices in different subgraphs is an external edge, and the two adjacent vertices are called out-neighbors to each other. By contrast, an edge joining vertices in the same subgraph is called an internal edges, and the two adjacent vertices are called in-neighbors to each other. Clearly, every vertex of has in-neighbors and two out-neighbors. For example, Fig. 1 depicts and , where each part of shadows in indicates a subgraph isomorphic to .

Fig. 1: (a) Alternating group graphs and .

Cheng et al. [14] propose the Split-star networks as alternatives to the star graphs and companion graphs with the alternating group graphs.

Definition 2.

(see [14]) The -dimensional split-star, denoted by , is a graph consisting of the vertex set and two vertices are adjacent if and only if or for some . That is, with .

In the above definition, the edge generated by the operation is called a -exchange edge, and others are called -rotation edges. Let be the set of all vertices in with the rightmost symbol , i.e., , . Also, let denote the subgraph of induced by . Clearly, the set forms a partition of and is isomorphic to . It is similar to that every vertex has two out-neighbors, which are joined to by external edges. Let and be subgraphs of

induced by the sets of even permutations and odd permutation, respectively, in which the adjacency applied to each subgraph is precisely using the edge of

-rotation. Clearly, is the alternating group graph , and is isomorphic via a mapping defined by -exchange. Accordingly, there are edges between and , called matching edges. Fig. 2 depicts , where dashed lines indicate matching edges.

Fig. 2: Split-star .

An independent set of a graph is a subset such that any two vertices of are nonadjacent in . For , we define , i.e., the set of neighbors of . Moreover, for , we define . When the graph is clear from the context, the subscript in the above notations are omitted. In what follows, we present some useful properties of , which will be adopted later.

2.1 Alternating group graphs and their properties

Lemma 2.1.

(see [25]) For with , the following properties hold:

(1) There are external edges between any two distinct subgraphs and for and .

(2) The two out-neighbors of every vertex of are contained in different subgraphs.

(3) If are two nonadjacent vertices of , then .

Lemma 2.2.

(see [16]) Let be a vertex-cut of with . If , then one of the following conditions holds:

(1) has two components, one of which is a singleton (i.e., a trivial component).

(2) has two components, one of which is an edge, say . In particular, .

Also, if , the above description still holds except for the following two exceptions. In both cases has two components, one of which is a -cycle and the other is either a -cycle (if ) or a -path (if ).

For example, and are two exceptions of described in Lemma 2.2, respectively (see Fig. 1). A graph is said to be hyper-connected [25, 31] or tightly super-connected [1] if each minimum vertex-cut creates exactly two components, one of which is a singleton. Since , the first exception illustrates that is not hyper-connected. Here we point out a minor flaw in the literatures (e.g., see Proposition 2.4 in [25] and Lemma 1 in [31]), which misrepresents that is hyper-connected. As a matter of fact, is isomorphic to the line graph of (i.e., a 3-dimensional hypercube), and the latter is contained in a list of vertex- and edge-transitive graphs without hyper-connectivity characterized by Meng [32]. For , since , by Lemma 2.2, is hyper-connected.

The following results are extensions of Lemma 2.2.

Lemma 2.3.

(see [15]) For , if is a vertex-cut of with , then one of the following conditions holds:

(1) has two components, one of which is a singleton or an edge.

(2) has three components, two of which are singletons.

Lemma 2.4.

(see [25]) For , if is a vertex-cut of with , then one of the following conditions holds:

(1) has two components, one of which is a singleton, an edge or a -path.

(2) has three components, two of which are singletons.

Lemma 2.5.

(see [31]) For , if is a vertex-cut of with , then one of the following conditions holds:

(1) has two components, one of which is a singleton, an edge, a -path or a -cycle.

(2) has three components, two of which are singletons or a singleton and an edge.

(3) has four components, three of which are singletons.

Lemma 2.6.

Let be an independent set of for . Then the following assertions hold.

(1) If , then .

(2) If , then .

Proof. Since is vertex-transitive, one may choose the identity permutation, denoted by , as a vertex in . Since is -regular, if (resp., ) and there exists no common neighbor between any two vertices of , then (resp., ), as required. In what follows, we assume that and let and . Clearly, and every vertex in has the symbol or at the last position. We further define

,

.

Since , the two sets and are identical. If , then has the symbol at the first position and symbol at the second position. In this case, we have , which meets the upper bound of Lemma 2.1(3) (see Fig. 3(a) for an illustration).

Claim 1.

For any two distinct vertices , . Moreover, if , then .

Proof of Claim 1. Let and . Consider the following situations: (i) and . In this case, if there exists a common neighbor, say , of and , then . Thus, (see, e.g., , and in Fig. 3(a)); (ii) and . In this case, if there exists a common neighbor, say , of and , then . Thus, (see, e.g., , and in Fig. 3(a)); (iii) and . In this case, it is clear that (see, e.g., and in Fig. 3(a)). This settles Claim 1.

On the other hand, the two sets and are not identical. Since every vertex in N(e) has two neighbors in and no two vertices of N(e) share a common neighbor, if , then . In fact, every vertex in has the symbol 1 at the first position, and every vertex in has the symbol 2 at the second position. Thus, both and are independent sets. Since the two symbols 1 and 2 are fixed in the first two positions for vertices in and respectively, every vertex in can be adjacent to at most one vertex of , and vice versa (see Fig. 3(b) for an illustration).

Claim 2.

For any two distinct vertices or , .

Proof of Claim 2. Without loss of generality, we consider . Let and . Consider the following situations: (i) and . In this case, if there exists a common neighbor, say , of and , then . Thus, (see, e.g., , and in Fig. 3(b)); (ii) and . In this case, if there exists a common neighbor, say , of and , then (see, e.g., , and in Fig. 3(b)); (iii) and . In this case, it is clear that (see, e.g., and in Fig. 3(b)). This settles Claim 2.

Note that two vertices and may have two common neighbors (see, e.g., and in Fig. 3(b). Then ).

Claim 3.

If and , either and are adjacent or .

Proof of Claim 3. Without loss of generality, we consider . Let and . Consider the following situations: (i) and . In this case, we have , and thus and are adjacent. (ii) and . In this case, if there exists a common neighbor, say , of and , then . Thus, (see, e.g., , and in Fig. 3); (iii) . In this case, it is clear that . This settles Claim 3.

Fig. 3: Illustration of Lemma 2.6, where each operation or is attached to an edge between vertices (from left to right).

We are now ready to conclude the proof of the lemma. Let and for any tow vertices . Consider the following conditions:

For (1), let . Since , at least one vertex for belongs to the sets . If , then . Since by Lemma 2.1(3), it implies . If , then . By Claim 1, we have . Thus, . If and (resp., and ), by Claim 3 either and are adjacent, which contradicts that is an independent set, or . Since and , it follows that . Therefore, we have for all above situations. Also, it is clear that if or , then .

For (2), let . Since , at least one vertex for belongs to the sets . Let and . If , then for and for and (by Claim 1). Thus, . If and , we have , , (by Claim 1), and (by Claim 3). Thus, . If and (resp., and ), we have , , (by Claim 2), and (by Claim 3). Thus, . If , and , we have , , (by Lemma 2.1(3)), and (by Claim 3). Thus, . If (resp., ), then for and for and (by Claim 2). Thus, . If and (resp., and ), we have for , (by Claim 2), and (by Lemma 2.1(3)). Thus, . Therefore, we have for all above situations. Also, if for any , by Case 1, we have .

Form Fig. 1 it easy to check that the set (resp., ) is an independent set of such that . Clearly, these examples show that the bounds on the assertions of Lemma 2.6 are tight for . Indeed, based on this observation, the following properties can easily be proved by induction on .

Remark 2.1.

For , the following assertions hold:

(1) The set for and is an independent set such that .

(2) The set for and is an independent set such that .

2.2 Split-stars and their properties

Lemma 2.7.

(see [11, 14, 13]) For with , the following properties hold:

  1. is -regular and for .

  2. The two out-neighbors of every vertex in are contained in different subgraphs and these two out-neighbors are adjacent. For any two vertices in the same subgraph , their out-neighbors in other subgraphs are different. There are external edges between any two distinct subgraphs and for and .

  3. If are any two vertices of , then

    where stands for the the distance (i.e., the number of edges in a shortest path) between and in .

Lemma 2.8.

(see [11]) For , if is a vertex-cut of with , then one of the following conditions holds:

(1) has two components, one of which is a singleton.

(2) has two components, one of which is an edge, say . If is a -exchange edge, then ; otherwise, , where , , and .

(3) has three components, two of which are singletons, say and . Moreover, and , hence .

Lemma 2.9.

(see [29]) For , if is a vertex-cut of with , then one of the following conditions holds:

(1) has two components, one of which is a singleton, an edge or a -path.

(2) has three components, two of which are singletons.

Lemma 2.10.

(see [29]) For , if is a vertex-cut of with , then one of the following conditions holds:

(1) has two components, one of which is a singleton, an edge, a -path or a -cycle.

(2) has three components, two of which are singletons or a singleton and an edge.

(3) has four components, three of which are singletons.

Lemma 2.11.

Let be an independent set of for . Then the following assertions hold.

(1) If , then .

(2) If , then .

(3) If , then .

Proof. Recall that contains two copies of , namely and . For notational convenience, we simply write , and as , and for any subset of vertices , respectively. Consider the following conditions:

For (1), let . By Lemma 2.7(3), and has at most two common neighbors,