The 4-Component Connectivity of Alternating Group Networks

08/19/2018 ∙ by Jou-Ming Chang, et al. ∙ 0

The ℓ-component connectivity (or ℓ-connectivity for short) 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 generalization is a natural extension of the classical connectivity defined in term of minimum vertex-cut. As an application, the ℓ-connectivity can be used to assess the vulnerability of a graph corresponding to the underlying topology of an interconnection network, and thus is an important issue for reliability and fault tolerance of the network. So far, only a little knowledge of results have been known on ℓ-connectivity for particular classes of graphs and small ℓ's. In a previous work, we studied the ℓ-connectivity on n-dimensional alternating group networks AN_n and obtained the result κ_3(AN_n)=2n-3 for n≥ 4. In this sequel, we continue the work and show that κ_4(AN_n)=3n-6 for n≥ 4.

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1 Introduction

As usual, the underlying topology of an interconnection network is modeled by a connected graph , where is the set of processors and is the set of communication links between processors. A subgraph obtained from by removing a set of vertices is denoted by . A separating set (or vertex-cut) of a connected graph is a set of vertices whose removal renders to become disconnected. If is not a complete graph, the connectivity is the cardinality of a minimum separating set of . By convention, the connectivity of a complete graph with vertices is defined to be . A graph is -connected if .

The connectivity is an important topic in graph theory. In particular, it plays a key role in applications related to the modern interconnection networks, e.g., can be used to assess the vulnerability of the corresponding network, and is an important measurement for reliability and fault tolerance of the network [28]. However, to further analyze the detailed situation of the disconnected network caused by a separating set, it is natural to generalize the classical connectivity by introducing some conditions or restrictions on the separating set and/or the components of  [14]. The most basic consideration is the number of components associated with the disconnected network. To figure out what kind of separating sets and/or how many sizes of a separating set can result in a disconnected network with a certain number of components, Chartrand et al. [5] proposed a generalization of connectivity with respect to separating set for making a more thorough study. In this paper, we follow this direction to investigate such kind of generalized connectivity on a class of interconnection networks called alternating group networks (defined later in Section 2).

For an integer , the generalized -connectivity of a graph , denoted by , is the minimum number of vertices whose removal from results in a disconnected graph with at least components or a graph with fewer than vertices. A graph is -connected if . A synonym for such a generalization was also called the general connectivity by Sampathkumar [26] or -component connectivity (-connectivity for short) by Hsu et al. [18], Cheng et al. [7, 8, 9] and Zhao et al. [29]. Hereafter, we follow the use of the terminology of Hsu et al. Obviously, . Similarly, for an integer , the generalized -edge-connectivity (-edge-connectivity for short) , which was introduced by Boesch and Chen [3], is defined to be the smallest number of edges whose removal leaves a graph with at least components if , and if . In addition, many problems related to networks on faulty edges haven been considered in [15, 16, 17, 25].

The notion of -connectivity is concerned with the relevance of the cardinality of a minimum vertex-cut and the number of components caused by the vertex-cut, which is a good measure of robustness of interconnection networks. Accordingly, this generalization is called the cut-version definition of generalized connectivity. We note that there are other diverse generalizations of connectivity in the literature, e.g., Hager [12] gave the so-called path-version definition of generalized connectivity, which is defined from the view point of Menger’s Theorem. Recently, Sun and Li [27] gave sharp bounds of the difference between the two versions of generalized connectivities.

For research results on -connectivity of graphs, the reader can refer to [5, 10, 11, 7, 8, 9, 18, 23, 24, 26, 29]. At the early stage, the main work focused on establishing sufficient conditions for graphs to be -connected, (e.g., see [5, 26, 23]). Also, several sharp bounds of -connectivity related to other graph parameters can be found in [26, 11]. In addition, for a graph and an integer , a function called -connectivity function is defined to be the minimum -edge-connectivity among all subgraphs of obtained by removing vertices from , and several properties of this function was investigated in [10, 24]. By contrast, finding -connectivity for certain interconnection networks is a new trend of research at present. So far, the exact values of -connectivity are known only for a few classes of networks, in particular, only for small ’s. For example, is determined on the -dimensional hypercube for (see [18]) and (see [29]), the -dimensional hierarchical cubic network (see [7]), the -dimensional complete cubic network (see [8]), and the generalized exchanged hypercube for and (see [9]). However, determining -connectivity is still unsolved for most interconnection networks. As a matter of fact, it has been pointed out in [18] that, unlike the hypercube, the results of the well-known interconnection networks such as the star graphs [1] and the alternating group graphs [20] are still unknown.

Recently, we studied two types of generalized 3-connectivities (i.e., the cut-version and the path-version of the generalized connectivities as mentioned before) in the -dimensional alternating group network , which was introduced by Ji [19] to serve as an interconnection network topology for computing systems. In [4], we already determined the 3-component connectivity for . In this sequel, we continue the work and show the following result.

Theorem 1.

For , .

2 Background of alternating group networks

Let and denote the set of all even permutations over . For , the -dimensional alternating group network, denoted by , is a graph with the vertex set of even permutations (i.e., ), and two vertices and are adjacent if and only if one of the following three conditions holds [19]:

(i) , , , and for .

(ii) , , , and for .

(iii) There exists an such that , , , , and for .

The basic properties of are known as follows. contains vertices and edges, which is a vertex-symmetric and -regular graph with diameter and connectivity . For and , let be the subnetwork of induced by vertices with the rightmost symbol in its permutation. It is clear that is isomorphic to . In fact, has a recursive structure, which can be constructed from disjoint copies for such that, for any two subnetworks and , and , there exist edges between them. Fig. 1 depicts , where each part of shadows indicates a subnetwork isomorphic to .

Fig. 1: Alternating group network .

A path (resp., cycle) of length is called a -path (resp., -cycle). For notational convenience, if a vertex belongs to a subnetwork , we simply write instead of . The disjoint union of two subnetworks and is denoted by . The subgraph obtained from by removing a set of vertices is denoted by . An edge with two end vertices and for is called an external edges between and . In this case, and are called out-neighbors to each other. By contrast, edges joining vertices in the same subnetwork are called internal edges, and the two adjacent vertices are called in-neighbors to each other. By definition, it is easy to check that every vertex of has in-neighbors and exactly one out-neighbor. Hereafter, for a vertex , we use to denote the set of in-neighbors of , and the unique out-neighbor of . Moreover, if is a subgraph of , we define as the in-neighborhood of , i.e., the set composed of all in-neighbors of those vertices in except for those belong to .

In what follow, we shall present some properties of , which will be used later. For more properties on alternating group networks, we refer to [6, 13, 19, 30, 31].

Lemma 1.

(see [13, 30, 31]) For with and with , the following holds:

(1) has no -cycle and -cycle.

(2) Any two distinct vertices of have different out-neighbors in .

(3) There are exactly edges between and .

Lemma 2.

For and , let be a connected induced subgraph of . Then, the following properties hold:

(1) If , then is a -cycle or a -path. Moreover, if is a -cycle (resp., a -path), then (resp., ).

(2) If , then .

Proof. The two properties can easily be proved by induction on . Now, we only verify the subgraph in Fig. 1 for the basis case . Recall that every vertex has in-neighbors in . For (1), the result of 3-cycle is clear. If is a 2-path, at most two adjacent vertices in can share a common in-neighbor, it follows the . For (2), the condition means that the number of vertices in cannot exceed a half of those in . In particular, if , then is either a claw (i.e., ), a paw (i.e., plus an edge), or a -path. Moreover, if is a paw, a claw or a 3-path, then no two adjacent vertices, at most one pair of adjacent vertices, or at most two pair of adjacent vertices in can share a common in-neighbor, respectively. This shows that when is a paw, when is a claw, and when is a 3-path. Also, if , it is clear that .

For designing a reliable probabilistic network, Bauer et al. [2] first introduced the notion of super connectedness. A regular graph is (loosely) super-connected if its only minimum vertex-cuts are those induced by the neighbors of a vertex, i.e., a minimum vertex-cut is the set of neighbors of a single vertex. If, in addition, the deletion of a minimum vertex-cut results in a graph with two components and one of which is a singleton, then the graph is tightly super-connected. More accurately, a graph is tightly -super-connected provided it is tightly super-connected and the cardinality of a minimum vertex-cut is equal to . Zhou and Xiao [30] pointed out that and are not super-connected, and showed that for is tightly -super-connected. Moreover, to evaluate the size of the connected components of with a set of faulty vertices, Zhou and Xiao gave the following properties.

Lemma 3.

(see [30]) For , if is a vertex-cut of with , then one of the following conditions holds:

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

(2) has two components, one of which is an edge, say . In particular, if , is composed of all neighbors of and , excluding and .

Lemma 4.

(see [30]) For , if is a vertex-cut of with , then one of the following conditions holds:

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

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

Through a more detailed analysis, Chang et al. [4] recently obtained a slight extension of the result of Lemma 3 as follows.

Lemma 5.

(see [4]) Let is a vertex-cut of with . Then, the following conditions hold:

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

(2) If , then has two components, one of which is a singleton, an edge, or a -cycle.

(3) If , then has two components, one of which is either a singleton or an edge.

3 The 4-component connectivity of

Since is a 3-cycle, by definition, it is clear that . Also, in the process of the drawing of Fig. 1, we found by a brute-force checking that the removal of no more than five vertices in (resp., eight vertices in ) results in a graph that is either connected or contains at most three components. Thus, the following lemma establishes the lower bound of for .

Lemma 6.

and .

Lemma 7.

For , .

Proof. Let be any vertex-cut in such that . For convenience, vertices in (resp., not in ) are called faulty vertices (resp., fault-free vertices). By Lemma 4, if , then contains at most three components. To complete the proof, we need to show that the same result holds for . Let and for each . We claim that there exists some subnetwork, say , such that it contains faulty vertices. Since , if it is so, then there are at most two such subnetworks. Suppose not, i.e., every subnetwork for has faulty vertices. Since is -connected, remains connected for each . Recall the property (3) of Lemma 1 that there are independent edges between and for each pair with . Since for , it guarantees that the two subgraphs and are connected by an external edge in . Thus, is connected, and this contradicts to the fact that is a vertex-cut in . Moreover, for such subnetworks, it is sure that some of must be a vertex-cut of . Otherwise, is connected, a contradiction. We now consider the following two cases:

Case 1: There is exactly one such subnetwork, say , such that it contains faulty vertices. In this case, we have for all and is a vertex-cut of . Let be the subgraph of induced by the fault-free vertices outside , i.e., . Since every subnetwork in has faulty vertices, from the previous argument it is sure that is connected. We denote by the component of that contains as its subgraph, and let be the number of faulty vertices outside . Since , we have . Consider the following scenarios:

Case 1.1: . In this case, there are no faulty vertices outside . That is, . Indeed, this case is impossible because if it is the case, then every vertex of has the fault-free out-neighbor in . Thus, belongs to , and it follows that is connected, a contradiction.

Case 1.2: . Let be the unique faulty vertex outside . That is, . Since is a vertex-cut of , we assume that is divided into disjoint connected components, say . For each , if , then there is at least one vertex of with its out-neighbor in , and thus belongs to . We now consider a component that is a singleton, say . If , then must be contained in , and thus belongs to . Clearly, there exists at most one component such that . In this case, has exactly two components and .

Case 1.3: . Let be the two faulty vertices outside . That is, . Since is a vertex-cut of , we assume that is divided into disjoint connected components, say . For each , if , then there is at least one vertex of with its out-neighbor in , and thus belongs to . We now consider a component with , i.e., is an edge, say . By the property (2) of Lemma 1, we have . If , then at least one of and must be contained in , and thus belongs to . Since and has in-neighbors (not including and ) in , we have for . Thus, there exists at most one such component such that . If it is the case of existence, then has exactly two components and . Finally, we consider a component that is a singleton. Since and every vertex has degree in , we have for . Thus, at most two such components exist in , say and where . If , then both and must be contained in , and thus and belong to . Also, if either or , then has exactly two components, one of which is a singleton or . Finally, if , then has exactly three components, two of which are singletons and .

Case 1.4: . Let be the three faulty vertices outside . That is, . Since is a vertex-cut of , we assume that is divided into disjoint connected components, say . For each , if , then there is at least one vertex of with its out-neighbor in , and thus belongs to . We now consider a component with , i.e., is either a 3-cycle or a 2-path. Assume that . If there is a vertex for , then must be contained in , and thus belongs to . Since and, by Lemma 2, we have , it follows that there exists at most one such component such that . If it is the case of existence, then has exactly two components, one of which is either a 3-cycle or a 2-path. Next, we consider a component with , i.e., is an edge, say . From an argument similar to Case 1.3 for analyzing the membership of and in the set , we can show that has exactly two components and . Finally, we consider a component that is a singleton. Then, an argument similar to Case 1.3 for analyzing singleton components shows that at most two such components exist in . Thus, has either two components (where one of which is a singleton) or three components (where two of which are singletons).

Case 1.5: . Let be the four faulty vertices outside . That is, . Since is a vertex-cut of , we assume that is divided into disjoint connected components, say . For each , if , then there is at least one vertex of with its out-neighbor in , and thus belongs to . If , by Lemma 2, we have . Since , it follows that for . Thus, none of component with exists in . Next, we consider a component with and assume . By Lemma 2, we have . Since is no more than , at most one such component exists in . Furthermore, if such exists, then it is either a 3-cycle or a 2-path. Thus, an argument similar to Case 1.4 for analyzing the membership of , and in the set , we can show that has exactly two components, one of which is a 3-cycle or a 2-path. Finally, if we consider a component with , an argument similar to the previous cases shows that has either two components (where one of which is a singleton or an edge) or three components (where two of which are singletons).

Case 1.6: . Let be the five faulty vertices outside . That is, . Since is a vertex-cut of , we assume that is divided into disjoint connected components, say . For each , if , then there is at least one vertex of with its out-neighbor in , and thus belongs to . If or , by Lemma 2, we have . Since , it follows that for . Thus, none of component with or exists in . We now consider a component with . Since , by Lemma 2, if such exists, then it must be a 3-cycle, and thus an argument similar to the previous cases shows that has exactly two components, one of which is a 3-cycle. Finally, if we consider a component with , an argument similar to the previous cases shows that has either two components (where one of which is a singleton or an edge) or three components (where two of which are singletons).

Case 1.7: . In this case, we have . Since is isomorphic to and is a vertex-cut of with no more than vertices, by Lemma 4, has at most three components as follows:

Case 1.7.1: has two components, one of which is either a singleton or an edge. Let and be such two components for which . More precisely, for . Clearly, the above inequality indicates that there exist some vertices of such that their out-neighbors are contained in , even if all out-neighbors of vertices in are contained in . Thus, belongs to . Also, if there is a vertex with its out-neighbor in , then belongs to . Otherwise, has exactly two components, one of which is either a singleton or an edge.

Case 1.7.2: has three components, two of which are singletons. Let and be such three components for which and . Since for , there exist some vertices of such that their out-neighbors are contained in . This shows that belongs to . Since has three components, the out-neighbor of a vertex or cannot be contained in . Thus, has exactly three components, two of which are singletons.

Case 2: There exist exactly two subnetworks, say and , such that . Since is a vertex-cut of , at least one of the subgraphs and must be disconnected. Let be the subgraph of induced by the fault-free vertices outside , i.e., . Since , we have for all . The bound of implies that is connected, and it follows that is also connected. We denote by the component of that contains as its subgraph. Since , we consider the following scenarios:

Case 2.1: . Clearly, . Since we have assumed , it follows that and there exist no faulty vertices outside . That is, . Indeed, this case is impossible because if it is the case, then there exist a vertex of such that its out-neighbor is contained in . Thus, belongs to , and it follows that is connected, a contradiction.

Case 2.2: . Since , it implies . Since is isomorphic to and , by Lemma 5, if is disconnected, then it has exactly two component, one of which is either a singleton or an edge. Suppose , where and are disjoint connected components such that . More precisely, for , where the last term is the number of faulty vertices outside . Clearly, the above inequality indicates that there exist some vertices of such that their out-neighbors are contained in , even if all out-neighbors of vertices in are contained in . Thus, belongs to . Also, if there is a vertex of with its out-neighbor in , then belongs to . By contrast, we can show that belongs to by a similar way if it is connected. Thus, contains at most one component (which is either a singleton or an edge) such that this component is a subgraph of . Similarly, since , contains at most one component (which is either a singleton or an edge) such that this component is a subgraph of . Thus, there are at most three components in . We claim that cannot simultaneously contain both an edge and a singleton as components. Suppose not and, without loss of generality, assume and . Then, at least two out-neighbors of and are not contained in . Otherwise, produces a 4-cycle or 5-cycle, which contradicts to the property (1) of Lemma 1. Thus, the number of faulty vertices of requires at least , a contradiction. Similarly, we claim that cannot simultaneously contain two disjoint edges and as components. Suppose not. By an argument similar above, we can show that either has faulty vertices for or it contains a 4-cycle or 5-cycle. However, both the cases are not impossible. Consequently, if contains three component, then two of which are singletons, one is a vertex of and the other is of .

Case 2.3: . Clearly, . Since is isomorphic to and , it is tightly -super-connected. Also, since , if is a vertex-cut of , then it must be a minimum vertex-cut. Particularly, there are two components in , one of which is a singleton, say . That is, all in-neighbors of are faulty vertices (i.e., ). Otherwise, is connected and thus belongs to . On the other hand, we consider all situations of as follows. Clearly, if is connected, then it belongs to , and this further implies that must be disconnected. In this case, contains exactly two components, one of which is a singleton . We now consider the case that is not connected and claim that it has at most two disjoint connected components. Suppose not. Since is isomorphic to , by Lemma 5, the number of faulty vertices in is at least . Since , it follows that . Thus, this situation is a symmetry of Case 2.1 by considering the exchange of and , which leads to a contradiction. Suppose , where and are disjoint connected components such that . Since for , where the last term is the number of faulty vertices outside . Clearly, the above inequality indicates that there exist some vertices of such that their out-neighbors are contained in , even if all out-neighbors of vertices in are contained in . Thus, belongs to . Also, if there is a vertex of with its out-neighbor in , then belongs to . Otherwise, is a component of . By Lemma 2, since when , we have . Moreover, since when , if , then must be a 3-cycle. If , then is either a singleton or an edge. Note that if is a 3-cycle or an edge, then cannot contain the the singleton as its component. Otherwise, an argument similar to Case 2.2 shows that either has more than faulty vertices or produces a 4-cycle or 5-cycle, a contradiction.

From the proof of Lemma 7, we obtain the following result, which is an extension of Lemma 4.

Corollary 8.

For , if is a vertex-cut of with , then one of the following conditions holds:

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

(2)