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The 4-Component Connectivity of Alternating Group Networks

The 4-Component Connectivity of Alternating Group Networks The `-component connectivity (or `-connectivity for short) of a graph G, denoted by (G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least ` components or a graph with fewer than ` vertices. This generalization is a natural extension of the classical connectivity de ned in term of minimum vertex- cut. As an application, the `-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 `-connectivity for particular classes of graphs and small `'s. In a previous work, we studied the `-connectivity on n-dimensional alternating group networks AN and obtained the result  (AN ) = 2n 3 for n > 4. In this sequel, n 3 n we continue the work and show that  (AN ) = 3n 6 for n > 4. 4 n Keyword: Interconnection networks, Graph connectivity, Generalized graph connectivity, Component connectivity, Alternating group networks This research was partially supported by the grant MOST-107-2221-E-141-001-MY3 from the Ministry of Science and Technology, Taiwan. Corresponding author. Email: [email protected] arXiv:1808.06160v1 [cs.DM] 19 Aug 2018 1 Introduction As usual, the underlying topology of an interconnection network is modeled by a connected graph G = (V; E), where V (= V (G)) is the set of processors and E(= E(G)) is the set of communication links between processors. A subgraph obtained from G by removing a set F of vertices is denoted by GF . A separating set (or vertex-cut ) of a connected graph G is a set F of vertices whose removal renders G F to become disconnected. If G is not a complete graph, the connectivity (G) is the cardinality of a minimum separating set of G. By convention, the connectivity of a complete graph with n vertices is de ned to be n 1. A graph G is n-connected if (G) > n. 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., (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 F and/or the components of G F [14]. The most basic consideration is the number of components associated with the disconnected network. To gure 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 (de ned later in Section 2). For an integer ` > 2, the generalized `-connectivity of a graph G, denoted by  (G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least ` components or a graph with fewer than ` vertices. A graph G is (n; `)-connected if (G) > n. 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{9] and Zhao et al. [29]. Hereafter, we follow the use of the terminology of Hsu et al. Obviously,  (G) = (G). Similarly, for an integer ` > 2, the generalized `-edge-connectivity (`-edge-connectivity for short)  (G), which was introduced by Boesch and Chen [3], is de ned to be the smallest number of edges whose removal leaves a graph with at least ` components if jV (G)j > `, and  (G) = jV (G)j if jV (G)j < `. In addition, many problems related to networks on faulty edges haven been considered in [15{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 de nition 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 de nition of generalized connectivity, which is de ned from the view point of Menger's Theorem. Recently, Sun and Li [27] gave sharp bounds of the di erence between the two versions of generalized connectivities. 2 For research results on `-connectivity of graphs, the reader can refer to [5, 7{11, 18, 23, 24, 26, 29]. At the early stage, the main work focused on establishing sucient conditions for graphs to be (n; `)-connected, (e.g., see [5, 23, 26]). Also, several sharp bounds of `-connectivity related to other graph parameters can be found in [11, 26]. In addition, for a graph G and an integer k 2 [0;  (G)], a function called `-connectivity function is de ned to be the minimum `-edge- connectivity among all subgraphs of G obtained by removing k vertices from G, and several properties of this function was investigated in [10, 24]. By contrast, nding `-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,  (G) is determined on the n-dimensional hypercube for ` 2 [2; n + 1] (see [18]) and ` 2 [n + 2; 2n 4] (see [29]), the n-dimensional hierarchical cubic network (see [7]), the n-dimensional complete cubic network (see [8]), and the generalized exchanged hypercube GEH (s; t) for 1 6 s 6 t and ` 2 [2; s + 1] (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 n-dimensional alter- nating group network AN , which was introduced by Ji [19] to serve as an interconnection network topology for computing systems. In [4], we already determined the 3-component con- nectivity  (AN ) = 2n 3 for n > 4. In this sequel, we continue the work and show the 3 n following result. Theorem 1. For n > 4,  (AN ) = 3n 6. 4 n 2 Background of alternating group networks Let Z = f1; 2; : : : ; ng and A denote the set of all even permutations over Z . For n > 3, the n n n n-dimensional alternating group network, denoted by AN , is a graph with the vertex set of even permutations (i.e., V (AN ) = A ), and two vertices p = (p p  p ) and q = (q q  q ) n n 1 2 n 1 2 n are adjacent if and only if one of the following three conditions holds [19]: (i) p = q , p = q , p = q , and p = q for j 2 Z nf1; 2; 3g. 1 2 2 3 3 1 j j n (ii) p = q , p = q , p = q , and p = q for j 2 Z nf1; 2; 3g. 1 3 2 1 3 2 j j n (iii) There exists an i 2 f4; 5; : : : ; ng such that p = q , p = q , p = q , p = q , and p = q 1 2 2 1 3 i i 3 j j for j 2 Z nf1; 2; 3; ig. The basic properties of AN are known as follows. AN contains n!=2 vertices and n!(n1)=4 n n edges, which is a vertex-symmetric and (n 1)-regular graph with diameter d3n=2e 3 and connectivity n 1. For n > 3 and i 2 Z , let AN be the subnetwork of AN induced by n n vertices with the rightmost symbol i in its permutation. It is clear that AN is isomorphic to AN . In fact, AN has a recursive structure, which can be constructed from n disjoint copies n1 n i i AN for i 2 Z such that, for any two subnetworks AN and AN , i; j 2 Z and i 6= j, there n n n n n 3 43521 23451 35421 24531 45231 43152 34125 53412 31452 32415 13425 35142 24315 45312 14352 31245 54132 41532 42135 14235 54213 45123 35214 25413 51423 31524 32154 13254 42513 15324 24153 41253 51234 Fig. 1: Alternating group network AN . exist (n 2)!=2 edges between them. Fig. 1 depicts AN , where each part of shadows indicates a subnetwork isomorphic to AN . A path (resp., cycle) of length k is called a k-path (resp., k-cycle ). For notational con- i i venience, if a vertex x belongs to a subnetwork AN , we simply write x 2 AN instead of n n j j i i i x 2 V (AN ). The disjoint union of two subnetworks AN and AN is denoted by AN [ AN . n n n n n The subgraph obtained from AN by removing a set F of vertices is denoted by AN F . n n An edge (x; y) 2 E(AN ) with two end vertices x 2 AN and y 2 AN for i 6= j is called an n n external edges between AN and AN . In this case, x and y 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 de nition, it is easy to check that every vertex of AN has n 2 in-neighbors and exactly one out-neighbor. Hereafter, for a vertex x 2 AN , we use N (x) to denote the set of in-neighbors of x, and out(x) the unique out- 4 S neighbor of x. Moreover, if H is a subgraph of AN , we de ne N (H ) = ( N (x))nV (H ) x2V (H) as the in-neighborhood of H , i.e., the set composed of all in-neighbors of those vertices in H except for those belong to H . In what follow, we shall present some properties of AN , 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 AN with n > 4 and i; j 2 Z with i 6= j, the following holds: n n (1) AN has no 4-cycle and 5-cycle. i i (2) Any two distinct vertices of AN have di erent out-neighbors in AN V (AN ). n n (3) There are exactly (n 2)!=2 edges between AN and AN . Lemma 2. For n > 6 and i 2 Z , let H be a connected induced subgraph of AN . Then, the following properties hold: (1) If jV (H )j = 3, then H is a 3-cycle or a 2-path. Moreover, if H is a 3-cycle (resp., a 2-path), then jN (H )j = 3n 12 (resp., 3n 11 6 jN (H )j 6 3n 10). (2) If 4 6 jV (H )j < (n 1)!=4, then jN (H )j > 4n 16. Proof. The two properties can easily be proved by induction on n. Now, we only verify the subgraph H in Fig. 1 for the basis case n = 6. Recall that every vertex has n 2 in-neighbors in AN . For (1), the result of 3-cycle is clear. If H is a 2-path, at most two adjacent vertices in H can share a common in-neighbor, it follows the 3n 11 6 jN (H )j 6 3n 10. For (2), the condition jV (H )j < (n 1)!=4 means that the number of vertices in H cannot exceed a half of those in AN . In particular, if jV (H )j = 4, then H is either a claw (i.e., K ), a paw (i.e., K 1;3 1;3 plus an edge), or a 3-path. Moreover, if H 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 H can share a common in-neighbor, respectively. This shows that jN (H )j = 4n 16 when H is a paw, 4n 15 6 jN (H )j 6 4n 14 when H is a claw, and 4n 16 6 jN (H )j 6 4n 14 when H is a 3-path. Also, if 4 < jV (H )j < (n 1)!=4, it is clear that jN (H )j > 4n 16. For designing a reliable probabilistic network, Bauer et al. [2] rst 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 k-super-connected provided it is tightly super-connected and the cardinality of a minimum vertex-cut is equal to k. Zhou and Xiao [30] pointed out that AN and AN are not super-connected, and showed that AN for n > 5 is 3 4 n tightly (n 1)-super-connected. Moreover, to evaluate the size of the connected components of AN with a set of faulty vertices, Zhou and Xiao gave the following properties. Lemma 3. (see [30]) For n > 5, if F is a vertex-cut of AN with jFj 6 2n 5, then one of the following conditions holds: (1) AN F has two components, one of which is a trivial component (i.e., a singleton). (2) AN F has two components, one of which is an edge, say (u; v). In particular, if jFj = 2n 5, F is composed of all neighbors of u and v, excluding u and v. 5 Lemma 4. (see [30]) For n > 5, if F is a vertex-cut of AN with jFj 6 3n 10, then one of the following conditions holds: (1) AN F has two components, one of which is either a singleton or an edge. (2) AN F 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 F is a vertex-cut of AN with jFj 6 2n 4. Then, the following conditions hold: (1) If n = 4, then AN F has two components, one of which is a singleton, an edge, a 3-cycle, a 2-path, or a paw. (2) If n = 5, then AN F has two components, one of which is a singleton, an edge, or a 3-cycle. (3) If n > 6, then AN F has two components, one of which is either a singleton or an edge. 3 The 4-component connectivity of AN Since AN is a 3-cycle, by de nition, it is clear that  (AN ) = 1. Also, in the process of the 3 4 3 drawing of Fig. 1, we found by a brute-force checking that the removal of no more than ve vertices in AN (resp., eight vertices in AN ) results in a graph that is either connected or 4 5 contains at most three components. Thus, the following lemma establishes the lower bound of (AN ) for n = 4; 5. 4 n Lemma 6.  (AN ) > 6 and  (AN ) > 9. 4 4 4 5 Lemma 7. For n > 6,  (AN ) > 3n 6. 4 n Proof. Let F be any vertex-cut in AN such that jFj 6 3n 7. For convenience, vertices in F (resp., not in F ) are called faulty vertices (resp., fault-free vertices). By Lemma 4, if jFj 6 3n 10, then AN F contains at most three components. To complete the proof, we need to show that the same result holds for 3n 9 6 jFj 6 3n 7. Let F = F \ V (AN ) and f = jF j for each i 2 Z . We claim that there exists some subnetwork, say AN , such that i i n it contains f > n 2 faulty vertices. Since 3(n 2) > 3n 7 > jFj, if it is so, then there are at most two such subnetworks. Suppose not, i.e., every subnetwork AN for j 2 Z has j j f 6 n 3 faulty vertices. Since AN is (n 2)-connected, AN F remains connected for n n j j each j 2 Z . Recall the property (3) of Lemma 1 that there are (n 2)!=2 independent edges between AN and AN for each pair i; j 2 Z with i 6= j. Since (n 2)!=2 > 2(n 3) > f + f n n i j for n > 6, it guarantees that the two subgraphs AN F and AN F are connected by an i n j external edge in AN F . Thus, AN F is connected, and this contradicts to the fact that n n F is a vertex-cut in AN . Moreover, for such subnetworks, it is sure that some of F must be n i a vertex-cut of AN . Otherwise, AN F is connected, a contradiction. We now consider the following two cases: 6 i Case 1: There is exactly one such subnetwork, say AN , such that it contains f > n 2 faulty vertices. In this case, we have f 6 n 3 for all j 2 Z n fig and F is a vertex-cut j n i i i of AN . Let H be the subgraph of AN induced by the fault-free vertices outside AN , i.e., n n H = AN (V (AN )[ F ). Since every subnetwork AN in H has f 6 n 3 faulty vertices, n n j from the previous argument it is sure that H is connected. We denote by C the component of AN F that contains H as its subgraph, and let f = jFj f be the number of faulty vertices n i outside AN . Since 3n 7 > jFj > f > n 2, we have 0 6 f 6 2n 5. Consider the following scenarios: Case 1.1: f = 0. In this case, there are no faulty vertices outside AN . That is, H = AN V (AN ). Indeed, this case is impossible because if it is the case, then every vertex of i i AN F has the fault-free out-neighbor in H . Thus, AN F belongs to C , and it follows i i n n that AN F is connected, a contradiction. Case 1.2: f = 1. Let u 2 F n F be the unique faulty vertex outside AN . That is, i i i H = AN (V (AN ) [ fug). Since F is a vertex-cut of AN , we assume that AN F is n i i n n n divided into k disjoint connected components, say C ; C ; : : : ; C . For each j 2 Z , if jC j > 2, 1 2 k k j then there is at least one vertex of C with its out-neighbor in H , and thus C belongs to C . j j We now consider a component that is a singleton, say C = fvg. If out(v) 6= u, then out(v) must be contained in H , and thus C belongs to C . Clearly, there exists at most one component C = fvg such that out(v) = u. In this case, AN F has exactly two components fvg and C . j n Case 1.3: f = 2. Let u ; u 2 F n F be the two faulty vertices outside AN . That is, 1 2 i i i i H = AN (V (AN )[fu ; u g). Since F is a vertex-cut of AN , we assume that AN F is n 1 2 i i n n n divided into k disjoint connected components, say C ; C ; : : : ; C . For each j 2 Z , if jC j > 3, 1 2 k k j then there is at least one vertex of C with its out-neighbor in H , and thus C belongs to j j C . We now consider a component C with jC j = 2, i.e., C is an edge, say (v; w). By the j j j property (2) of Lemma 1, we have out(v) 6= out(w). If fout(v); out(w)g =6 fu ; u g, then at 1 2 least one of out(v) and out(w) must be contained in H , and thus C belongs to C . Since (3n 7) 2 > f = jFj f > (3n 9) 2 and (v; w) has 2n 6 in-neighbors (not including v and w) in AN , we have 2n 6 < f < 2(2n 6) for n > 6. Thus, there exists at most one such component C = f(v; w)g such that fout(v); out(w)g = fu ; u g. If it is the case of j 1 2 existence, then AN F has exactly two components f(v; w)g and C . Finally, we consider a component that is a singleton. Since 3n 9 6 f 6 3n 11 and every vertex has degree n 2 in AN , we have n 2 < f < 3(n 2) for n > 6. Thus, at most two such components exist i 0 in AN F , say C = fvg and C = fwg where j; j 2 Z . If out(v); out(w) 2= fu ; u g, then i j j k 1 2 both out(v) and out(w) must be contained in H , and thus C and C belong to C . Also, if j j either out(v) 2= fu ; u g or out(w) 2= fu ; u g, then AN F has exactly two components, one 1 2 1 2 n of which is a singleton fvg or fwg. Finally, if fout(v); out(w)g = fu ; u g, then AN F has 1 2 n exactly three components, two of which are singletons fvg and fwg. Case 1.4: f = 3. Let u ; u ; u 2 F n F be the three faulty vertices outside AN . That is, 1 2 3 i i i i H = AN (V (AN )[fu ; u ; u g). Since F is a vertex-cut of AN , we assume that AN F n 1 2 3 i i n n n is divided into k disjoint connected components, say C ; C ; : : : ; C . For each j 2 Z , if jC j > 4, 1 2 k k j then there is at least one vertex of C with its out-neighbor in H , and thus C belongs to C . We j j now consider a component C with jC j = 3, i.e., C is either a 3-cycle or a 2-path. Assume that j j j 7 V (C ) = fv ; v ; v g. If there is a vertex out(v ) 2= fu ; u ; u g for 1 6 h 6 3, then out(v ) must j 1 2 3 h 1 2 3 h be contained in H , and thus C belongs to C . Since (3n7)3 > f = jFjf > (3n9)3 and, j i by Lemma 2, we have 3n 12 6 jN (C )j 6 n 10, it follows that there exists at most one such component C such that fout(v ); out(v ); out(v )g = fu ; u ; u g. If it is the case of existence, j 1 2 3 1 2 3 then AN F has exactly two components, one of which is either a 3-cycle or a 2-path. Next, we consider a component C with jC j = 2, i.e., C is an edge, say (v; w). From an argument j j j similar to Case 1.3 for analyzing the membership of out(v) and out(w) in the set fu ; u ; u g, 1 2 3 we can show that AN F has exactly two components f(v; w)g and C . 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 AN F . Thus, AN F has i n either two components (where one of which is a singleton) or three components (where two of which are singletons). Case 1.5: f = 4. Let u ; u ; u ; u 2 F n F be the four faulty vertices outside AN . That 1 2 3 4 i i i is, H = AN (V (AN ) [ fu ; u ; u ; u g). Since F is a vertex-cut of AN , we assume that n 1 2 3 4 i n n AN F is divided into k disjoint connected components, say C ; C ; : : : ; C . For each j 2 Z , if i 1 2 k k jC j > 5, then there is at least one vertex of C with its out-neighbor in H , and thus C belongs j j j to C . If jC j > 4, by Lemma 2, we have jN (C )j > 4n 16. Since (3n 7) 4 > jFj f = f , j j i it follows that jN (C )j > f for n > 6. Thus, none of component C with jC j = 4 exists in j i j j AN . Next, we consider a component C with jC j = 3 and assume V (C ) = fv ; v ; v g. By j j j 1 2 3 Lemma 2, we have 3n 12 6 jN (C )j 6 n 10. Since f is no more than 3n 11, at most j i one such component C exists in AN F . Furthermore, if such C exists, then it is either a j i j 3-cycle or a 2-path. Thus, an argument similar to Case 1.4 for analyzing the membership of out(v ), out(v ) and out(v ) in the set fu ; u ; u ; u g, we can show that AN F has exactly 1 2 3 1 2 3 4 n two components, one of which is a 3-cycle or a 2-path. Finally, if we consider a component C with jC j 6 2, an argument similar to the previous cases shows that AN F has either two j n components (where one of which is a singleton or an edge) or three components (where two of which are singletons). Case 1.6: f = 5. Let u ; u ; u ; u ; u 2 F n F be the ve faulty vertices outside AN . 1 2 3 4 5 i i i That is, H = AN (V (AN )[fu ; u ; u ; u ; u g). Since F is a vertex-cut of AN , we assume n 1 2 3 4 5 i n n that AN F is divided into k disjoint connected components, say C ; C ; : : : ; C . For each i 1 2 n k j 2 Z , if jC j > 6, then there is at least one vertex of C with its out-neighbor in H , and thus j j C belongs to C . If jC j = 4 or jC j = 5, by Lemma 2, we have jN (C )j > 4n 16. Since j j j j (3n 7) 5 > jFj f = f , it follows that jN (C )j > f for n > 6. Thus, none of component i j i C with jC j = 4 or jC j = 5 exists in AN . We now consider a component C with jC j = 3. j j j j j Since f 6 3n 12, by Lemma 2, if such C exists, then it must be a 3-cycle, and thus an i j argument similar to the previous cases shows that AN F has exactly two components, one of which is a 3-cycle. Finally, if we consider a component C with jC j 6 2, an argument similar j j to the previous cases shows that AN F 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: 6 6 f 6 2n5. In this case, we have (3n7)6 > f = jFjf > (3n9)(2n5). i i Since AN is isomorphic to AN and F is a vertex-cut of AN with no more than 3(n1)10 n1 i n n vertices, by Lemma 4, AN F has at most three components as follows: 8 i Case 1.7.1: AN F has two components, one of which is either a singleton or an edge. Let C and C be such two components for which 1 6 jC j 6 2 < jC j. More precisely, 1 2 1 2 jC j = jV (AN )j f jC j > (n 1)!=2 f 2 > (3n 7) f > jFj f = f for n > 6. 2 i 1 i i i Clearly, the above inequality indicates that there exist some vertices of C such that their out- neighbors are contained in H , even if all out-neighbors of vertices in F n F are contained in C . Thus, C belongs to C . Also, if there is a vertex v 2 C with its out-neighbor in H , then 2 2 1 C belongs to C . Otherwise, AN F has exactly two components, one of which is either a 1 n singleton or an edge. Case 1.7.2: AN F has three components, two of which are singletons. Let C ; C i 1 2 and C be such three components for which jC j = jC j = 1 and jC j > 2. Since jC j = 3 1 2 3 3 (n 1)!=2 f 2 > (3n 7) f > jFj f = f for n > 6, there exist some vertices of C such i i i 2 that their out-neighbors are contained in H . This shows that C belongs to C . Since AN F 2 i has three components, the out-neighbor of a vertex v 2 C or v 2 C cannot be contained in 1 2 H . Thus, AN F has exactly three components, two of which are singletons. Case 2: There exist exactly two subnetworks, say AN and AN , such that f ; f > n 2. n i j Since F is a vertex-cut of AN , at least one of the subgraphs AN F and AN F must n i n j be disconnected. Let H be the subgraph of AN induced by the fault-free vertices outside j j i i AN [ AN , i.e., H = AN (V (AN )[ V (AN )[ F ). Since 2n 4 6 f + f 6 jFj 6 3n 7, n n n i j n n we have f 6 jFj f f 6 (3n 7) (2n 4) = n 3 for all h 2 Z nfi; jg. The bound of f h i j n h implies that AN F is connected, and it follows that H is also connected. We denote by C the component of AN F that contains H as its subgraph. Since n 2 6 f 6 (3n 7)f 6 n i j (3n 7) (n 2) = 2n 5, we consider the following scenarios: Case 2.1: f = 2n 5. Clearly, f 6 (3n 7) f = n 2. Since we have assumed i j i f > n 2, it follows that f = n 2 and there exist no faulty vertices outside AN [ AN . j j n That is, H = AN (V (AN )[ V (AN )). Indeed, this case is impossible because if it is the case, then there exist a vertex of (AN [AN )F such that its out-neighbor is contained in H . Thus, (AN [AN )F belongs to C , and it follows that AN F is connected, a contradiction. Case 2.2: n 1 6 f 6 2n 6. Since f + f 6 jFj 6 3n 7, it implies f 6 (3n 7) f 6 i i j j i (3n 7) (n 1) = 2n 6. Since AN is isomorphic to AN and f 6 2(n 1) 4, by n1 i Lemma 5, if AN F is disconnected, then it has exactly two component, one of which is either a singleton or an edge. Suppose AN F = C [ C , where C and C are disjoint connected i 1 2 1 2 components such that 1 6 jC j 6 2 < jC j. More precisely, jC j = jV (AN )j f jC j = 1 2 2 i 1 (n 1)!=2 f 2 > (3n 7) f > jFj f for n > 6, where the last term jFj f is the i i i i number of faulty vertices outside AN . Clearly, the above inequality indicates that there exist some vertices of C such that their out-neighbors are contained in H , even if all out-neighbors of vertices in FnF are contained in C . Thus, C belongs to C . Also, if there is a vertex of C with i 2 2 1 its out-neighbor in H , then C belongs to C . By contrast, we can show that AN F belongs 1 i to C by a similar way if it is connected. Thus, AN F contains at most one component (which is either a singleton or an edge) such that this component is a subgraph of AN . Similarly, since f 6 2n 6, AN F contains at most one component (which is either a singleton j n or an edge) such that this component is a subgraph of AN . Thus, there are at most three components in AN F . We claim that AN F cannot simultaneously contain both an edge n n 9 (u; v) and a singleton w as components. Suppose not and, without loss of generality, assume u; v 2 AN and w 2 AN . Then, at least two out-neighbors of u; v and w are not contained in N (u) [ N (v) [ N (w). Otherwise, AN produces a 4-cycle or 5-cycle, which contradicts to the property (1) of Lemma 1. Thus, the number of faulty vertices of AN requires at least (2n 6) + (n 2) + 2 = 3n 6 > jFj, a contradiction. Similarly, we claim that AN F cannot simultaneously contain two disjoint edges (u ; v ) and (u ; v ) as components. Suppose not. By 1 1 2 2 an argument similar above, we can show that either AN has 2(2n 6) + 2 > 3n 7 > jFj faulty vertices for n > 6 or it contains a 4-cycle or 5-cycle. However, both the cases are not impossible. Consequently, if AN F contains three component, then two of which are singletons, one is a vertex of AN and the other is of AN . Case 2.3: f = n 2. Clearly, f 6 (3n 7) f = 2n 5. Since AN is isomorphic to i j i AN and n > 6, it is tightly (n 2)-super-connected. Also, since f = n 2, if F is a vertex- n1 i i cut of AN , then it must be a minimum vertex-cut. Particularly, there are two components in AN F , one of which is a singleton, say v. That is, all in-neighbors of v are faulty vertices (i.e., N (v) = F ). Otherwise, AN F is connected and thus belongs to C . On the other hand, we i i j j consider all situations of AN F as follows. Clearly, if AN F is connected, then it belongs n j n j to C , and this further implies that AN F must be disconnected. In this case, AN F i n contains exactly two components, one of which is a singleton v. We now consider the case that AN F is not connected and claim that it has at most two disjoint connected components. n j Suppose not. Since AN is isomorphic to AN , by Lemma 5, the number of faulty vertices in n n1 AN is at least 2(n 1) 3. Since f 6 2n 5, it follows that f = 2n 5. Thus, this situation is n j j a symmetry of Case 2.1 by considering the exchange of f and f , which leads to a contradiction. i j Suppose AN F = C [ C , where C and C are disjoint connected components such that n j 1 2 1 2 jC j 6 jC j. Since jC j > (jV (AN )j f )=2 > (n 1)!=4 f > (3n 7) f > jFj f for 1 2 2 n j j j j n > 6, where the last term jFj f is the number of faulty vertices outside AN . Clearly, the j n above inequality indicates that there exist some vertices of C such that their out-neighbors are contained in H , even if all out-neighbors of vertices in F n F are contained in C . Thus, C j 2 2 belongs to C . Also, if there is a vertex of C with its out-neighbor in H , then C belongs to C . 1 1 Otherwise, C is a component of AN F . By Lemma 2, since f = 2n 5 < 4n 16 when 1 n j n > 6, we have jC j < 4. Moreover, since 2n 5 6 3n 11 when n > 6, if jC j = 3, then 1 1 C must be a 3-cycle. If jC j 6 2, then C is either a singleton or an edge. Note that if C is 1 1 1 1 a 3-cycle or an edge, then AN F cannot contain the the singleton v 2 V (AN F ) as its n i component. Otherwise, an argument similar to Case 2.2 shows that AN either has more than 3n 7 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 n > 5, if F is a vertex-cut of AN with jFj 6 3n 7, then one of the following conditions holds: (1) AN F has two components, one of which is either a singleton, an edge, a 3-cycle, or a 2-path. (2) AN F has three components, two of which are singletons. Proof of Theorem 1. Lemmas 6 and 7 show that  (AN ) > 3n 6 for n > 4. To complete 4 n 10 the proof, we need to show the upper bound  (AN ) 6 3n 6 for n > 4. Consider an induced 4 n 6-cycle H = (v ; v ; v ; v ; v ; v ) in AN (the existence of such a cycle can be veri ed in Fig 1). 1 2 3 4 5 6 n Let F be the set composed of all neighbors of vertices in fv ; v ; v g. Since every vertex of AN 1 3 5 n has n 1 neighbors and any two vertices in fv ; v ; v g share a common neighbor, it is clear 1 3 5 that jFj = 3(n 1) 3 = 3n 6. Then, the removal of F from AN leads to the surviving graph with a large connected component and three singletons v ; v and v . 1 3 5 4 Concluding remarks In this paper, we follow a previous work to investigate a measure of network reliability, called `-component connectivity, in alternating group networks AN . Although we have known that (AN ) = 2n 3 and  (AN ) = 3n 6, at this stage it remains open to determine  (AN ) 3 n 4 n n for ` > 5. Also, as aforementioned, by now little work has been done in determining the `- component connectivity for most interconnection networks, even if for smaller integer `. In the future work, we would like to study the `-component connectivity of AN with larger `, or some popular interconnection networks such as star graphs (and their super class of graphs called (n; k)-star graphs and arrangement graphs), bubble-sort graphs, and alternating group graphs. References [1] S.B. Akers, D. Harel, and B. Krishnamurthy, The star graph: An attractive alternative to the n-cube, in: Proc. Int. Conf. Parallel Processing (ICPP'1987), University Park, August 1987, pp. 393{400. [2] D. Bauer, F. Boesch, C. Su el, and R. Tindell, Connectivity extremal problems and the design of reliable probabilistic networks, The Theory and Application of Graphs, Y. Alavi and G. Chartrand (Editors), Wiley, New York, 1981, pp. 89{98. [3] F.T. Boesch, S. 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Hennayake, H.-J. Lai, D. Li, J. Mao, Minimally (k; k)-edge-connected graphs, J. Graph Theory 44 (2003) 116{131. [16] S.-Y. Hsieh, C.-H. Chen, Pancyclicity on M obius cubes with maximal edge faults, Parallel Comput. 30 (2004) 407{421. [17] S.-Y. Hsieh, G.-H. Chen, C.-W. Ho, Longest fault-free paths in star graphs with edge faults, IEEE Trans. Comput. 50 (2001) 960{971. [18] L.-H. Hsu, E. Cheng, L. Lipt ak, J.J.M. Tan, C.-K. Lin, T.-Y. Ho, Component connectivity of the hypercubes, Int. J. Comput. Math. 89 (2012) 137{145. [19] Y.-H. Ji, A class of Cayley networks based on the alternating groups, Adv. Math. 4 (1998) 361{362. (in Chinese) [20] J. Jwo, S. Lakshmivarahan, S.K. Dhall, A new class of interconnection networks based on the alternating group, Networks 23 (1993) 315{326. [21] X. Li, Y. Mao, Generalized Connectivity of Graphs, Springer Briefs in Mathematics, Springer, New York, 2016. [22] X. Li, Y. Mao, A survey on the generalized connectivity of graphs, arXiv:1207.1838v9 (2014). [23] O.R. Oellermann, On the l-connectivity of a graph, Graph Comb. 3 (1987) 285{291. [24] O.R. Oellermann, A note on the l-connectivity function of a graph, Cong. Num. 60 (1987) 181{188. [25] X. Pan, J. Mao, H. Liu, Minimally (k; k 1)-edge-connected graphs, Australas. J. Comb. 28 (2003) 39{49. [26] E. Sampathkumar, Connectivity of a graph { A generalization, J. Combin. Inform. Sys. Sci. 9 (1984) 71{78. [27] Y. Sun, X. Li, On the di erence of two generalized connectivities of a graph, J. Comb. Optim. 33 (2017) 283{291. [28] J.M. Xu, Topological Structure and Analysis of Interconnection Networks, Kluwer Aca- demic Publishers, London, 2001. [29] S. Zhao, W. Yang, S. Zhang, Component connectivity of hypercubes, Theor. Comput. Sci. 640 (2016) 115{118. [30] S. Zhou, W. Xiao, Conditional diagnosability of alternating group networks, Inform. Pro- cess. Lett. 110 (2010) 403{409. [31] S. Zhou, W. Xiao, B. Parhami, Construction of vertex-disjoint paths in alternating group networks, J. 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The 4-Component Connectivity of Alternating Group Networks

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Abstract

The `-component connectivity (or `-connectivity for short) of a graph G, denoted by (G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least ` components or a graph with fewer than ` vertices. This generalization is a natural extension of the classical connectivity de ned in term of minimum vertex- cut. As an application, the `-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 `-connectivity for particular classes of graphs and small `'s. In a previous work, we studied the `-connectivity on n-dimensional alternating group networks AN and obtained the result  (AN ) = 2n 3 for n > 4. In this sequel, n 3 n we continue the work and show that  (AN ) = 3n 6 for n > 4. 4 n Keyword: Interconnection networks, Graph connectivity, Generalized graph connectivity, Component connectivity, Alternating group networks This research was partially supported by the grant MOST-107-2221-E-141-001-MY3 from the Ministry of Science and Technology, Taiwan. Corresponding author. Email: [email protected] arXiv:1808.06160v1 [cs.DM] 19 Aug 2018 1 Introduction As usual, the underlying topology of an interconnection network is modeled by a connected graph G = (V; E), where V (= V (G)) is the set of processors and E(= E(G)) is the set of communication links between processors. A subgraph obtained from G by removing a set F of vertices is denoted by GF . A separating set (or vertex-cut ) of a connected graph G is a set F of vertices whose removal renders G F to become disconnected. If G is not a complete graph, the connectivity (G) is the cardinality of a minimum separating set of G. By convention, the connectivity of a complete graph with n vertices is de ned to be n 1. A graph G is n-connected if (G) > n. 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., (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 F and/or the components of G F [14]. The most basic consideration is the number of components associated with the disconnected network. To gure 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 (de ned later in Section 2). For an integer ` > 2, the generalized `-connectivity of a graph G, denoted by  (G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least ` components or a graph with fewer than ` vertices. A graph G is (n; `)-connected if (G) > n. 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{9] and Zhao et al. [29]. Hereafter, we follow the use of the terminology of Hsu et al. Obviously,  (G) = (G). Similarly, for an integer ` > 2, the generalized `-edge-connectivity (`-edge-connectivity for short)  (G), which was introduced by Boesch and Chen [3], is de ned to be the smallest number of edges whose removal leaves a graph with at least ` components if jV (G)j > `, and  (G) = jV (G)j if jV (G)j < `. In addition, many problems related to networks on faulty edges haven been considered in [15{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 de nition 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 de nition of generalized connectivity, which is de ned from the view point of Menger's Theorem. Recently, Sun and Li [27] gave sharp bounds of the di erence between the two versions of generalized connectivities. 2 For research results on `-connectivity of graphs, the reader can refer to [5, 7{11, 18, 23, 24, 26, 29]. At the early stage, the main work focused on establishing sucient conditions for graphs to be (n; `)-connected, (e.g., see [5, 23, 26]). Also, several sharp bounds of `-connectivity related to other graph parameters can be found in [11, 26]. In addition, for a graph G and an integer k 2 [0;  (G)], a function called `-connectivity function is de ned to be the minimum `-edge- connectivity among all subgraphs of G obtained by removing k vertices from G, and several properties of this function was investigated in [10, 24]. By contrast, nding `-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,  (G) is determined on the n-dimensional hypercube for ` 2 [2; n + 1] (see [18]) and ` 2 [n + 2; 2n 4] (see [29]), the n-dimensional hierarchical cubic network (see [7]), the n-dimensional complete cubic network (see [8]), and the generalized exchanged hypercube GEH (s; t) for 1 6 s 6 t and ` 2 [2; s + 1] (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 n-dimensional alter- nating group network AN , which was introduced by Ji [19] to serve as an interconnection network topology for computing systems. In [4], we already determined the 3-component con- nectivity  (AN ) = 2n 3 for n > 4. In this sequel, we continue the work and show the 3 n following result. Theorem 1. For n > 4,  (AN ) = 3n 6. 4 n 2 Background of alternating group networks Let Z = f1; 2; : : : ; ng and A denote the set of all even permutations over Z . For n > 3, the n n n n-dimensional alternating group network, denoted by AN , is a graph with the vertex set of even permutations (i.e., V (AN ) = A ), and two vertices p = (p p  p ) and q = (q q  q ) n n 1 2 n 1 2 n are adjacent if and only if one of the following three conditions holds [19]: (i) p = q , p = q , p = q , and p = q for j 2 Z nf1; 2; 3g. 1 2 2 3 3 1 j j n (ii) p = q , p = q , p = q , and p = q for j 2 Z nf1; 2; 3g. 1 3 2 1 3 2 j j n (iii) There exists an i 2 f4; 5; : : : ; ng such that p = q , p = q , p = q , p = q , and p = q 1 2 2 1 3 i i 3 j j for j 2 Z nf1; 2; 3; ig. The basic properties of AN are known as follows. AN contains n!=2 vertices and n!(n1)=4 n n edges, which is a vertex-symmetric and (n 1)-regular graph with diameter d3n=2e 3 and connectivity n 1. For n > 3 and i 2 Z , let AN be the subnetwork of AN induced by n n vertices with the rightmost symbol i in its permutation. It is clear that AN is isomorphic to AN . In fact, AN has a recursive structure, which can be constructed from n disjoint copies n1 n i i AN for i 2 Z such that, for any two subnetworks AN and AN , i; j 2 Z and i 6= j, there n n n n n 3 43521 23451 35421 24531 45231 43152 34125 53412 31452 32415 13425 35142 24315 45312 14352 31245 54132 41532 42135 14235 54213 45123 35214 25413 51423 31524 32154 13254 42513 15324 24153 41253 51234 Fig. 1: Alternating group network AN . exist (n 2)!=2 edges between them. Fig. 1 depicts AN , where each part of shadows indicates a subnetwork isomorphic to AN . A path (resp., cycle) of length k is called a k-path (resp., k-cycle ). For notational con- i i venience, if a vertex x belongs to a subnetwork AN , we simply write x 2 AN instead of n n j j i i i x 2 V (AN ). The disjoint union of two subnetworks AN and AN is denoted by AN [ AN . n n n n n The subgraph obtained from AN by removing a set F of vertices is denoted by AN F . n n An edge (x; y) 2 E(AN ) with two end vertices x 2 AN and y 2 AN for i 6= j is called an n n external edges between AN and AN . In this case, x and y 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 de nition, it is easy to check that every vertex of AN has n 2 in-neighbors and exactly one out-neighbor. Hereafter, for a vertex x 2 AN , we use N (x) to denote the set of in-neighbors of x, and out(x) the unique out- 4 S neighbor of x. Moreover, if H is a subgraph of AN , we de ne N (H ) = ( N (x))nV (H ) x2V (H) as the in-neighborhood of H , i.e., the set composed of all in-neighbors of those vertices in H except for those belong to H . In what follow, we shall present some properties of AN , 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 AN with n > 4 and i; j 2 Z with i 6= j, the following holds: n n (1) AN has no 4-cycle and 5-cycle. i i (2) Any two distinct vertices of AN have di erent out-neighbors in AN V (AN ). n n (3) There are exactly (n 2)!=2 edges between AN and AN . Lemma 2. For n > 6 and i 2 Z , let H be a connected induced subgraph of AN . Then, the following properties hold: (1) If jV (H )j = 3, then H is a 3-cycle or a 2-path. Moreover, if H is a 3-cycle (resp., a 2-path), then jN (H )j = 3n 12 (resp., 3n 11 6 jN (H )j 6 3n 10). (2) If 4 6 jV (H )j < (n 1)!=4, then jN (H )j > 4n 16. Proof. The two properties can easily be proved by induction on n. Now, we only verify the subgraph H in Fig. 1 for the basis case n = 6. Recall that every vertex has n 2 in-neighbors in AN . For (1), the result of 3-cycle is clear. If H is a 2-path, at most two adjacent vertices in H can share a common in-neighbor, it follows the 3n 11 6 jN (H )j 6 3n 10. For (2), the condition jV (H )j < (n 1)!=4 means that the number of vertices in H cannot exceed a half of those in AN . In particular, if jV (H )j = 4, then H is either a claw (i.e., K ), a paw (i.e., K 1;3 1;3 plus an edge), or a 3-path. Moreover, if H 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 H can share a common in-neighbor, respectively. This shows that jN (H )j = 4n 16 when H is a paw, 4n 15 6 jN (H )j 6 4n 14 when H is a claw, and 4n 16 6 jN (H )j 6 4n 14 when H is a 3-path. Also, if 4 < jV (H )j < (n 1)!=4, it is clear that jN (H )j > 4n 16. For designing a reliable probabilistic network, Bauer et al. [2] rst 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 k-super-connected provided it is tightly super-connected and the cardinality of a minimum vertex-cut is equal to k. Zhou and Xiao [30] pointed out that AN and AN are not super-connected, and showed that AN for n > 5 is 3 4 n tightly (n 1)-super-connected. Moreover, to evaluate the size of the connected components of AN with a set of faulty vertices, Zhou and Xiao gave the following properties. Lemma 3. (see [30]) For n > 5, if F is a vertex-cut of AN with jFj 6 2n 5, then one of the following conditions holds: (1) AN F has two components, one of which is a trivial component (i.e., a singleton). (2) AN F has two components, one of which is an edge, say (u; v). In particular, if jFj = 2n 5, F is composed of all neighbors of u and v, excluding u and v. 5 Lemma 4. (see [30]) For n > 5, if F is a vertex-cut of AN with jFj 6 3n 10, then one of the following conditions holds: (1) AN F has two components, one of which is either a singleton or an edge. (2) AN F 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 F is a vertex-cut of AN with jFj 6 2n 4. Then, the following conditions hold: (1) If n = 4, then AN F has two components, one of which is a singleton, an edge, a 3-cycle, a 2-path, or a paw. (2) If n = 5, then AN F has two components, one of which is a singleton, an edge, or a 3-cycle. (3) If n > 6, then AN F has two components, one of which is either a singleton or an edge. 3 The 4-component connectivity of AN Since AN is a 3-cycle, by de nition, it is clear that  (AN ) = 1. Also, in the process of the 3 4 3 drawing of Fig. 1, we found by a brute-force checking that the removal of no more than ve vertices in AN (resp., eight vertices in AN ) results in a graph that is either connected or 4 5 contains at most three components. Thus, the following lemma establishes the lower bound of (AN ) for n = 4; 5. 4 n Lemma 6.  (AN ) > 6 and  (AN ) > 9. 4 4 4 5 Lemma 7. For n > 6,  (AN ) > 3n 6. 4 n Proof. Let F be any vertex-cut in AN such that jFj 6 3n 7. For convenience, vertices in F (resp., not in F ) are called faulty vertices (resp., fault-free vertices). By Lemma 4, if jFj 6 3n 10, then AN F contains at most three components. To complete the proof, we need to show that the same result holds for 3n 9 6 jFj 6 3n 7. Let F = F \ V (AN ) and f = jF j for each i 2 Z . We claim that there exists some subnetwork, say AN , such that i i n it contains f > n 2 faulty vertices. Since 3(n 2) > 3n 7 > jFj, if it is so, then there are at most two such subnetworks. Suppose not, i.e., every subnetwork AN for j 2 Z has j j f 6 n 3 faulty vertices. Since AN is (n 2)-connected, AN F remains connected for n n j j each j 2 Z . Recall the property (3) of Lemma 1 that there are (n 2)!=2 independent edges between AN and AN for each pair i; j 2 Z with i 6= j. Since (n 2)!=2 > 2(n 3) > f + f n n i j for n > 6, it guarantees that the two subgraphs AN F and AN F are connected by an i n j external edge in AN F . Thus, AN F is connected, and this contradicts to the fact that n n F is a vertex-cut in AN . Moreover, for such subnetworks, it is sure that some of F must be n i a vertex-cut of AN . Otherwise, AN F is connected, a contradiction. We now consider the following two cases: 6 i Case 1: There is exactly one such subnetwork, say AN , such that it contains f > n 2 faulty vertices. In this case, we have f 6 n 3 for all j 2 Z n fig and F is a vertex-cut j n i i i of AN . Let H be the subgraph of AN induced by the fault-free vertices outside AN , i.e., n n H = AN (V (AN )[ F ). Since every subnetwork AN in H has f 6 n 3 faulty vertices, n n j from the previous argument it is sure that H is connected. We denote by C the component of AN F that contains H as its subgraph, and let f = jFj f be the number of faulty vertices n i outside AN . Since 3n 7 > jFj > f > n 2, we have 0 6 f 6 2n 5. Consider the following scenarios: Case 1.1: f = 0. In this case, there are no faulty vertices outside AN . That is, H = AN V (AN ). Indeed, this case is impossible because if it is the case, then every vertex of i i AN F has the fault-free out-neighbor in H . Thus, AN F belongs to C , and it follows i i n n that AN F is connected, a contradiction. Case 1.2: f = 1. Let u 2 F n F be the unique faulty vertex outside AN . That is, i i i H = AN (V (AN ) [ fug). Since F is a vertex-cut of AN , we assume that AN F is n i i n n n divided into k disjoint connected components, say C ; C ; : : : ; C . For each j 2 Z , if jC j > 2, 1 2 k k j then there is at least one vertex of C with its out-neighbor in H , and thus C belongs to C . j j We now consider a component that is a singleton, say C = fvg. If out(v) 6= u, then out(v) must be contained in H , and thus C belongs to C . Clearly, there exists at most one component C = fvg such that out(v) = u. In this case, AN F has exactly two components fvg and C . j n Case 1.3: f = 2. Let u ; u 2 F n F be the two faulty vertices outside AN . That is, 1 2 i i i i H = AN (V (AN )[fu ; u g). Since F is a vertex-cut of AN , we assume that AN F is n 1 2 i i n n n divided into k disjoint connected components, say C ; C ; : : : ; C . For each j 2 Z , if jC j > 3, 1 2 k k j then there is at least one vertex of C with its out-neighbor in H , and thus C belongs to j j C . We now consider a component C with jC j = 2, i.e., C is an edge, say (v; w). By the j j j property (2) of Lemma 1, we have out(v) 6= out(w). If fout(v); out(w)g =6 fu ; u g, then at 1 2 least one of out(v) and out(w) must be contained in H , and thus C belongs to C . Since (3n 7) 2 > f = jFj f > (3n 9) 2 and (v; w) has 2n 6 in-neighbors (not including v and w) in AN , we have 2n 6 < f < 2(2n 6) for n > 6. Thus, there exists at most one such component C = f(v; w)g such that fout(v); out(w)g = fu ; u g. If it is the case of j 1 2 existence, then AN F has exactly two components f(v; w)g and C . Finally, we consider a component that is a singleton. Since 3n 9 6 f 6 3n 11 and every vertex has degree n 2 in AN , we have n 2 < f < 3(n 2) for n > 6. Thus, at most two such components exist i 0 in AN F , say C = fvg and C = fwg where j; j 2 Z . If out(v); out(w) 2= fu ; u g, then i j j k 1 2 both out(v) and out(w) must be contained in H , and thus C and C belong to C . Also, if j j either out(v) 2= fu ; u g or out(w) 2= fu ; u g, then AN F has exactly two components, one 1 2 1 2 n of which is a singleton fvg or fwg. Finally, if fout(v); out(w)g = fu ; u g, then AN F has 1 2 n exactly three components, two of which are singletons fvg and fwg. Case 1.4: f = 3. Let u ; u ; u 2 F n F be the three faulty vertices outside AN . That is, 1 2 3 i i i i H = AN (V (AN )[fu ; u ; u g). Since F is a vertex-cut of AN , we assume that AN F n 1 2 3 i i n n n is divided into k disjoint connected components, say C ; C ; : : : ; C . For each j 2 Z , if jC j > 4, 1 2 k k j then there is at least one vertex of C with its out-neighbor in H , and thus C belongs to C . We j j now consider a component C with jC j = 3, i.e., C is either a 3-cycle or a 2-path. Assume that j j j 7 V (C ) = fv ; v ; v g. If there is a vertex out(v ) 2= fu ; u ; u g for 1 6 h 6 3, then out(v ) must j 1 2 3 h 1 2 3 h be contained in H , and thus C belongs to C . Since (3n7)3 > f = jFjf > (3n9)3 and, j i by Lemma 2, we have 3n 12 6 jN (C )j 6 n 10, it follows that there exists at most one such component C such that fout(v ); out(v ); out(v )g = fu ; u ; u g. If it is the case of existence, j 1 2 3 1 2 3 then AN F has exactly two components, one of which is either a 3-cycle or a 2-path. Next, we consider a component C with jC j = 2, i.e., C is an edge, say (v; w). From an argument j j j similar to Case 1.3 for analyzing the membership of out(v) and out(w) in the set fu ; u ; u g, 1 2 3 we can show that AN F has exactly two components f(v; w)g and C . 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 AN F . Thus, AN F has i n either two components (where one of which is a singleton) or three components (where two of which are singletons). Case 1.5: f = 4. Let u ; u ; u ; u 2 F n F be the four faulty vertices outside AN . That 1 2 3 4 i i i is, H = AN (V (AN ) [ fu ; u ; u ; u g). Since F is a vertex-cut of AN , we assume that n 1 2 3 4 i n n AN F is divided into k disjoint connected components, say C ; C ; : : : ; C . For each j 2 Z , if i 1 2 k k jC j > 5, then there is at least one vertex of C with its out-neighbor in H , and thus C belongs j j j to C . If jC j > 4, by Lemma 2, we have jN (C )j > 4n 16. Since (3n 7) 4 > jFj f = f , j j i it follows that jN (C )j > f for n > 6. Thus, none of component C with jC j = 4 exists in j i j j AN . Next, we consider a component C with jC j = 3 and assume V (C ) = fv ; v ; v g. By j j j 1 2 3 Lemma 2, we have 3n 12 6 jN (C )j 6 n 10. Since f is no more than 3n 11, at most j i one such component C exists in AN F . Furthermore, if such C exists, then it is either a j i j 3-cycle or a 2-path. Thus, an argument similar to Case 1.4 for analyzing the membership of out(v ), out(v ) and out(v ) in the set fu ; u ; u ; u g, we can show that AN F has exactly 1 2 3 1 2 3 4 n two components, one of which is a 3-cycle or a 2-path. Finally, if we consider a component C with jC j 6 2, an argument similar to the previous cases shows that AN F has either two j n components (where one of which is a singleton or an edge) or three components (where two of which are singletons). Case 1.6: f = 5. Let u ; u ; u ; u ; u 2 F n F be the ve faulty vertices outside AN . 1 2 3 4 5 i i i That is, H = AN (V (AN )[fu ; u ; u ; u ; u g). Since F is a vertex-cut of AN , we assume n 1 2 3 4 5 i n n that AN F is divided into k disjoint connected components, say C ; C ; : : : ; C . For each i 1 2 n k j 2 Z , if jC j > 6, then there is at least one vertex of C with its out-neighbor in H , and thus j j C belongs to C . If jC j = 4 or jC j = 5, by Lemma 2, we have jN (C )j > 4n 16. Since j j j j (3n 7) 5 > jFj f = f , it follows that jN (C )j > f for n > 6. Thus, none of component i j i C with jC j = 4 or jC j = 5 exists in AN . We now consider a component C with jC j = 3. j j j j j Since f 6 3n 12, by Lemma 2, if such C exists, then it must be a 3-cycle, and thus an i j argument similar to the previous cases shows that AN F has exactly two components, one of which is a 3-cycle. Finally, if we consider a component C with jC j 6 2, an argument similar j j to the previous cases shows that AN F 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: 6 6 f 6 2n5. In this case, we have (3n7)6 > f = jFjf > (3n9)(2n5). i i Since AN is isomorphic to AN and F is a vertex-cut of AN with no more than 3(n1)10 n1 i n n vertices, by Lemma 4, AN F has at most three components as follows: 8 i Case 1.7.1: AN F has two components, one of which is either a singleton or an edge. Let C and C be such two components for which 1 6 jC j 6 2 < jC j. More precisely, 1 2 1 2 jC j = jV (AN )j f jC j > (n 1)!=2 f 2 > (3n 7) f > jFj f = f for n > 6. 2 i 1 i i i Clearly, the above inequality indicates that there exist some vertices of C such that their out- neighbors are contained in H , even if all out-neighbors of vertices in F n F are contained in C . Thus, C belongs to C . Also, if there is a vertex v 2 C with its out-neighbor in H , then 2 2 1 C belongs to C . Otherwise, AN F has exactly two components, one of which is either a 1 n singleton or an edge. Case 1.7.2: AN F has three components, two of which are singletons. Let C ; C i 1 2 and C be such three components for which jC j = jC j = 1 and jC j > 2. Since jC j = 3 1 2 3 3 (n 1)!=2 f 2 > (3n 7) f > jFj f = f for n > 6, there exist some vertices of C such i i i 2 that their out-neighbors are contained in H . This shows that C belongs to C . Since AN F 2 i has three components, the out-neighbor of a vertex v 2 C or v 2 C cannot be contained in 1 2 H . Thus, AN F has exactly three components, two of which are singletons. Case 2: There exist exactly two subnetworks, say AN and AN , such that f ; f > n 2. n i j Since F is a vertex-cut of AN , at least one of the subgraphs AN F and AN F must n i n j be disconnected. Let H be the subgraph of AN induced by the fault-free vertices outside j j i i AN [ AN , i.e., H = AN (V (AN )[ V (AN )[ F ). Since 2n 4 6 f + f 6 jFj 6 3n 7, n n n i j n n we have f 6 jFj f f 6 (3n 7) (2n 4) = n 3 for all h 2 Z nfi; jg. The bound of f h i j n h implies that AN F is connected, and it follows that H is also connected. We denote by C the component of AN F that contains H as its subgraph. Since n 2 6 f 6 (3n 7)f 6 n i j (3n 7) (n 2) = 2n 5, we consider the following scenarios: Case 2.1: f = 2n 5. Clearly, f 6 (3n 7) f = n 2. Since we have assumed i j i f > n 2, it follows that f = n 2 and there exist no faulty vertices outside AN [ AN . j j n That is, H = AN (V (AN )[ V (AN )). Indeed, this case is impossible because if it is the case, then there exist a vertex of (AN [AN )F such that its out-neighbor is contained in H . Thus, (AN [AN )F belongs to C , and it follows that AN F is connected, a contradiction. Case 2.2: n 1 6 f 6 2n 6. Since f + f 6 jFj 6 3n 7, it implies f 6 (3n 7) f 6 i i j j i (3n 7) (n 1) = 2n 6. Since AN is isomorphic to AN and f 6 2(n 1) 4, by n1 i Lemma 5, if AN F is disconnected, then it has exactly two component, one of which is either a singleton or an edge. Suppose AN F = C [ C , where C and C are disjoint connected i 1 2 1 2 components such that 1 6 jC j 6 2 < jC j. More precisely, jC j = jV (AN )j f jC j = 1 2 2 i 1 (n 1)!=2 f 2 > (3n 7) f > jFj f for n > 6, where the last term jFj f is the i i i i number of faulty vertices outside AN . Clearly, the above inequality indicates that there exist some vertices of C such that their out-neighbors are contained in H , even if all out-neighbors of vertices in FnF are contained in C . Thus, C belongs to C . Also, if there is a vertex of C with i 2 2 1 its out-neighbor in H , then C belongs to C . By contrast, we can show that AN F belongs 1 i to C by a similar way if it is connected. Thus, AN F contains at most one component (which is either a singleton or an edge) such that this component is a subgraph of AN . Similarly, since f 6 2n 6, AN F contains at most one component (which is either a singleton j n or an edge) such that this component is a subgraph of AN . Thus, there are at most three components in AN F . We claim that AN F cannot simultaneously contain both an edge n n 9 (u; v) and a singleton w as components. Suppose not and, without loss of generality, assume u; v 2 AN and w 2 AN . Then, at least two out-neighbors of u; v and w are not contained in N (u) [ N (v) [ N (w). Otherwise, AN produces a 4-cycle or 5-cycle, which contradicts to the property (1) of Lemma 1. Thus, the number of faulty vertices of AN requires at least (2n 6) + (n 2) + 2 = 3n 6 > jFj, a contradiction. Similarly, we claim that AN F cannot simultaneously contain two disjoint edges (u ; v ) and (u ; v ) as components. Suppose not. By 1 1 2 2 an argument similar above, we can show that either AN has 2(2n 6) + 2 > 3n 7 > jFj faulty vertices for n > 6 or it contains a 4-cycle or 5-cycle. However, both the cases are not impossible. Consequently, if AN F contains three component, then two of which are singletons, one is a vertex of AN and the other is of AN . Case 2.3: f = n 2. Clearly, f 6 (3n 7) f = 2n 5. Since AN is isomorphic to i j i AN and n > 6, it is tightly (n 2)-super-connected. Also, since f = n 2, if F is a vertex- n1 i i cut of AN , then it must be a minimum vertex-cut. Particularly, there are two components in AN F , one of which is a singleton, say v. That is, all in-neighbors of v are faulty vertices (i.e., N (v) = F ). Otherwise, AN F is connected and thus belongs to C . On the other hand, we i i j j consider all situations of AN F as follows. Clearly, if AN F is connected, then it belongs n j n j to C , and this further implies that AN F must be disconnected. In this case, AN F i n contains exactly two components, one of which is a singleton v. We now consider the case that AN F is not connected and claim that it has at most two disjoint connected components. n j Suppose not. Since AN is isomorphic to AN , by Lemma 5, the number of faulty vertices in n n1 AN is at least 2(n 1) 3. Since f 6 2n 5, it follows that f = 2n 5. Thus, this situation is n j j a symmetry of Case 2.1 by considering the exchange of f and f , which leads to a contradiction. i j Suppose AN F = C [ C , where C and C are disjoint connected components such that n j 1 2 1 2 jC j 6 jC j. Since jC j > (jV (AN )j f )=2 > (n 1)!=4 f > (3n 7) f > jFj f for 1 2 2 n j j j j n > 6, where the last term jFj f is the number of faulty vertices outside AN . Clearly, the j n above inequality indicates that there exist some vertices of C such that their out-neighbors are contained in H , even if all out-neighbors of vertices in F n F are contained in C . Thus, C j 2 2 belongs to C . Also, if there is a vertex of C with its out-neighbor in H , then C belongs to C . 1 1 Otherwise, C is a component of AN F . By Lemma 2, since f = 2n 5 < 4n 16 when 1 n j n > 6, we have jC j < 4. Moreover, since 2n 5 6 3n 11 when n > 6, if jC j = 3, then 1 1 C must be a 3-cycle. If jC j 6 2, then C is either a singleton or an edge. Note that if C is 1 1 1 1 a 3-cycle or an edge, then AN F cannot contain the the singleton v 2 V (AN F ) as its n i component. Otherwise, an argument similar to Case 2.2 shows that AN either has more than 3n 7 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 n > 5, if F is a vertex-cut of AN with jFj 6 3n 7, then one of the following conditions holds: (1) AN F has two components, one of which is either a singleton, an edge, a 3-cycle, or a 2-path. (2) AN F has three components, two of which are singletons. Proof of Theorem 1. Lemmas 6 and 7 show that  (AN ) > 3n 6 for n > 4. To complete 4 n 10 the proof, we need to show the upper bound  (AN ) 6 3n 6 for n > 4. Consider an induced 4 n 6-cycle H = (v ; v ; v ; v ; v ; v ) in AN (the existence of such a cycle can be veri ed in Fig 1). 1 2 3 4 5 6 n Let F be the set composed of all neighbors of vertices in fv ; v ; v g. Since every vertex of AN 1 3 5 n has n 1 neighbors and any two vertices in fv ; v ; v g share a common neighbor, it is clear 1 3 5 that jFj = 3(n 1) 3 = 3n 6. Then, the removal of F from AN leads to the surviving graph with a large connected component and three singletons v ; v and v . 1 3 5 4 Concluding remarks In this paper, we follow a previous work to investigate a measure of network reliability, called `-component connectivity, in alternating group networks AN . Although we have known that (AN ) = 2n 3 and  (AN ) = 3n 6, at this stage it remains open to determine  (AN ) 3 n 4 n n for ` > 5. 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Published: Aug 19, 2018

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