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The conditional fault diagnosability of (n, k)-star graphs
Applied Mathematics and Computation 218 (2012) 9742–9749
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The conditional fault diagnosability of (n, k)-star graphs Shuming Zhou Key Laboratory of Network Security and Cryptology, Fujian Normal University, Fuzhou, Fujian 350007, PR China College of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350007, PR China
a r t i c l e
i n f o
Keywords: Fault tolerance Comparison diagnosis Conditional diagnosability (n, k)-Star graphs
a b s t r a c t The growing size of the multiprocessor system increases its vulnerability to component failures. It is crucial to locate and replace the faulty processors to maintain a system’s high reliability. The fault diagnosis is the process of identifying faulty processors in a system through testing. This paper shows that the largest connected component of the survival graph contains almost all the remaining vertices when a lot of faulty vertices occur in the (n, k)-star graph Sn;k ðn P 5; k P 3; n k P 2Þ. Based on this fault resiliency, it establishes the conditional fault diagnosability of Sn;k under the comparison model. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Distributed processor architectures offer the potential advantage of high speed, provided that they are highly fault tolerant and reliable, and have good communication between remote processors. An important component of such a distributed system is its graph topology, which defines the inter-processor communication architecture. Fault tolerance can be especially important for interconnection networks, since computers may fail, creating faults in the network. To be reliable, the rest of the network should stay connected. Obviously, this can only be guaranteed if the number of faults is smaller than the smallest degree in the network. When the number of faults is larger than the smallest degree, some extensions of connectivity are necessary, since the graph may become disconnected. Thus, several types of connectivity were introduced and examined for various classes of graphs, such as loosely super connectedness and tightly super connectedness [1], super connectivity [2], super k-restricted edge connectivity [3]. As we increase the number of faults in the graph, it is desirable that most of the network stays connected, with at most a few processors separated from the rest, since then the network will continue to be able to function. Many interconnection networks have been examined in this aspect, when the number of faults is about twice the smallest degree [4,5]. One can even go further and ask what happens when more vertices are deleted. This was examined for the hypercube in [6–8] and for certain Cayley graphs generated by transpositions in [9], and it was shown that the resulting network will have a large component containing almost all vertices. The process of identifying faulty processors in a system by analyzing the outcomes of available inter-processor tests is called system-level diagnosis. The foundation of system diagnosis and an original diagnostic model, namely the PMC model, were established in a classic paper by Preparata et al. [10]. Its target is to identify the exact set of all faulty nodes before their repair or replacement. All tests are performed between two adjacent processors, and it was assumed that a test result is reliable (respectively, unreliable) if the processor that initiates the test is fault-free (respectively, faulty). The comparison-based diagnosis models, first proposed by Malek [11] and Chwa and Hakimi [12], have been considered to be a practical approach for fault diagnosis in multiprocessor systems. In these models, the same job is assigned to a pair of processors in the system and their outputs are compared by a central observer. This central observer performs diagnosis using the outcomes of these
⇑ Address: College of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350007, PR China. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.021
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comparisons. Maeng and Malek [13] extended Malek’s comparison approach to allow the comparisons carried out by the processors themselves. Sengupta and Dahbura [14] developed this comparison approach such that the comparisons have no central unit involved. Lin et al. [15] introduced the conditional diagnosis. By evaluating the size of connected components, they obtained that the conditional diagnosability of Star graph Sn is 3n 7 which is about three times larger than the classical diagnosability of star graphs. Using the evaluation of the size of connected components, Hsu et al. [16] have recently proved that the conditional diagnosability of hypercube is 3n 5. This idea was attributed to Lai et al. [17] who is the first to use a restricted diagnosis strategy. They obtained that the conditional diagnosability of the hypercubes Q n is 4n 7 under the PMC model. Furthermore, Hsu et al. [16] exposed the difference between these two conditional diagnosis models. This paper establishes the conditional diagnosability of Sn;k (n P 5; k P 3; n k P 2) under the comparison diagnosis model based on the fault tolerance of (n, k)-star graph Sn;k . The rest of this paper is organized as follows. Section 2 introduces some definitions, notations and the structure of (n, k)-star graph Sn;k . Section 3 is devoted to the fault resiliency of Sn;k ; and Section 4 concentrates on the conditional diagnosability of Sn;k . Section 5 concludes the paper. 2. Preliminaries Throughout this paper, we use a graph G ¼ GðV; EÞ to represent an interconnection network, where each node u 2 V denotes a processor and each edge ðu; v Þ 2 E denotes a link between nodes u and v. Let S be a subset of VðGÞ. The subgraph of G induced by S, denoted by G½S, is the graph with the vertex set S \ VðGÞ and the edge set fðu; v Þjðu; v Þ 2 EðGÞ; u; v 2 Sg. For any subset F V, the notation G n F (or G F) represents the graph obtained by removing the vertices in F from G and deleting those edges with at least one end vertex in F simultaneously. If G n F is disconnected, F is called a vertex cut or a separating set. The components of G n F are its maximal connected subgraphs. For any node u of G; NðuÞ denotes the set of all its neighS boring nodes, i.e., NðuÞ ¼ fv jðu; v Þ 2 Eg. For any set F V, let NðFÞ ¼ u2F NðuÞ F; N½F ¼ NðFÞ [ F. For brevity, N½u ¼ NðuÞ [ fug, Nðfu; v gÞ and N½fu; v g are written as Nðu; v Þ and N½u; v . The symmetric difference of any two sets F 1 and F 2 is defined as the set F 1 DF 2 ¼ ðF 1 F 2 Þ [ ðF 2 F 1 Þ. Network reliability is one of the major factors in designing the topology of an interconnection network. The hypercube and its variants were the first major class of interconnection networks. The n-star graph (Sn for short), which proposed by Akers and Krishnamurthy is an attractive alternative to the hypercube [18]. An Sn is a Cayley graph with a regular and hierarchical structure; for a similar number of vertices, the graph has a lower vertex degree, a smaller diameter, and a shorter average distance than the comparable hypercube [19]. However, with the restriction on the number of vertices n!, there is a large gap between n! and (n + 1)! for expanding an Sn to an Snþ1 . To relax the restriction of the numbers of vertices n! in Sn , a generalized version of the star graph, the (n, k)-star graph, was proposed by Chiang and Chen [20]. An Sn;k preserves many attractive properties of Sn such as vertex symmetry, hierarchical structure, maximal fault tolerance, and simple shortest routing. The two parameters n and k can be tuned to make a suitable choice for the number of vertices in the network and for the degree/diameter tradeoff. An n! Sn;k is regular of degree n 1, and the number of vertices is ðnkÞ! . They form a hierarchical family of graphs, each of which is node-symmetric; they can be recursively decomposed in a number of ways; they have a simple shortest-path routing algorithm [20]; the node-connectivity of Sn;k is n 1 [21]; there is an exact formula for their diameters, and their fault-diameters are at most their fault-free diameters plus 3 [21]; Hsu et al. [22] proved that Sn;k F is Hamiltonian (resp. Hamiltonian connected) if jFj 6 n 3 (resp. jFj 6 n 4) with F VðSn;k Þ [ EðSn;k Þ and n k P 2; Lin and Duh [23] also proved that the (n, k)-star graph Sn;k is super spanning connected if n > 3 and n k P 2. Chen et al. [24] successfully gave a method for constructing vertex-disjoint paths in (n, k)-star graphs. Chen et al. [24] showed that (n, k)-star graph Sn;k is 6-weak-vertex-pancyclic. Recently, Xiang and Stewart [25] presented an algorithm to find one-to-many node-disjoint paths in (n, k)-star graphs. Now, we formally present the structure of the (n, k)-star graph Sn;k . For simplicity, we use hni to denote the set f1; 2; . . . ; ng. The (n, k)-star graph, denoted by Sn;k , is a graph with the vertex set VðSn;k Þ ¼ fu1 u2 uk1 uk jui 2 hni, and ui – uj for i – jg, and adjacency is defined as follows: A vertex u1 u2 uk1 uk is adjacent to. (1) the vertex ui u2 ui1 u1 uiþ1 uk , where 2 6 i 6 k; (2) the vertex xu2 uk1 uk , where x 2 hni fu1 u2 uk1 uk g. The edges of type (1) are referred to as i-edges and the two end vertices are i-neighbors to each other. The edges of type (2) are referred to as 1-edges and, the two end vertices are 1-neighbors to each other [19]. The structure of an S4;2 is shown in Fig. 1. An Sn;k can be formed by interconnecting n Sn1;k1 ’s, that is, an Sn;k can be decomposed into Sn1;k1 ’s along any dimension i, and it can also be decomposed into n vertex disjoint Sn1;k1 ’s in k 1 different ways by fixing one symbol in any position i; 2 6 i 6 k. We denote Sin;k the subgraph which fixes the symbol i in the last position k. Obviously, Sin;k ffi Sn1;k1 . This decomposition can be recursively carried out on each Sin;k to obtain smaller subgraphs. The graph Sin;k is undirected and vertex-symmetric, but two different types of edges (1-edge and i-edge with 2 6 i 6 k) in the graph Sn;k prevent being edge symmetric. 3. Fault tolerance of (n, k)-star graph S n;k The connectivity jðGÞ of a graph G ¼ GðV; EÞ is the minimum number of nodes whose removal results in a disconnected or a trivial (one node) graph. The connectivity jðGÞ of G is an important parameter to measure the fault tolerance of the
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Fig. 1. S4,2.
network, while it has an obvious deficiency in that it tacitly assume that all elements in any subset of G can potentially fail at the same time. To compensate for this shortcoming, it would seem natural to generalize the classical connectivity by introducing some conditions or restrictions on the separating set S and/or the components of G n S. A k-regular graph is maximally connected if it is k-connected. A k-regular graph is (loosely) super k-connected if any one of its minimum separating sets is a set of the neighbors of some vertex. If, in addition, the deletion of a minimum separating set results in a graph with two components (one of which has only one vertex), then the graph is tightly super k-connected. Note that a k-regular graph can be loosely super connected but not tightly super connected [1]. Consequently, Esfahanian introduced the concepts of the restricted cut and the restricted connectivity of a graph [26]. A restricted vertex set S is a restricted vertex-cut if G n S is disconnected, and no component is an isolated vertex. The restricted vertex connectivity of a graph G, denoted by j0 ðGÞ, is the minimum cardinality of a restricted vertex-cut. It has been shown that if a network possesses the restricted connectivity property, it is more reliable and has the lower vertex failure comparing to that has only the super connectivity property. Usually, if the surviving graph contains a large connected component, it may be used as the functional subsystem, without incurring severe performance degradation. Thus, in evaluating a distributed system, it is indispensable to estimate the size of the maximal connected components of the underlying graph when the structure begins to lose processors. Yang et al. [6–8] proved that the hypercube Q n with a set F of at most 2n 3 failing processors has a component of size 2n jFj 1; and that it has a component of size 2n jFj 2 if jFj 6 3n 6. Similar result of Star graph Sn was also obtained by Yang et al. [27,28]. Cheng et al., [29,30] gave a more detail on this result as follows: Let F be a set of vertices of at most 2n 4, if Sn F is not connected, Sn F has two components, one of which has exactly one vertex, or has two vertices with a edge, and in the latter case, F is the union of the neighbor sets of these two vertices excluding themselves. Cheng and Lipták [9] generalized this result for star graph with linearly many faults. In this section, we discuss the fault resilience of Sn;k . Theorem 1. The (n, k)-star graph Sn;k ðn P 5; k P 3; n k P 2Þ is tightly (n 1)-super connected.
Proof. As the connectivity of the (n, k)-star graph Sn;k (n P 5; k P 3; n k P 2) is n 1, we suppose that F is a subset of P VðSn;k Þ with jFj ¼ n 1, and Sn;k n F is not connected. Let F i ¼ VðSin;k Þ \ F, and fi ¼ jF i j for 1 6 i 6 n. So ni¼1 fi ¼ jFj ¼ n 1. We discuss as follows: (1) fi 6 n 3 for 8i 2 f1; 2; . . . ; ng. For 8i 2 f1; 2; . . . ; ng; Sin;k n F i is connected since Sin;k is n 2-connected. If Sin;k n F i is connected to Sjn;k n F j for 0 8i – j 2 f1; 2; . . . ; ng; Sn;k n F is connected; otherwise, there exist i0 ; j0 2 f1; 2; . . . ; ng such that Sin;k n F i0 is not directly j0 i ðn2Þ! connected to Sn;k n F j0 . As there are ðnkÞ! matching edges between two subgraphs Sn;k and Sjn;k for 8i – j 2 f1; 2; . . . ; ng, and ðn2Þ! P n 2 P 3 for n P 5, k P 3; n k P 2, there are at least n 2 faulty vertices which destroy ðnkÞ! j0 0 the matching edges between Sin;k n F i0 and Sn;k n F j0 , while there is at most one faulty vertex which can destroy another j0 i0 matching edge, so Sn;k n F i0 can connect to Sn;k n F j0 indirectly through the help of some subgraph (s) of fS1n;k ; j0 0 S2n;k ; . . . ; Snn;k g n fSin;k ; Sn;k g. Thus Sn;k n F is connected, a contradiction. (2) There exists some subgraph, say Sin;k , which has fi ¼ n 1 faulty vertices, consequently, F j ¼ VðSjn;k Þ \ F ¼ / for 8j 2 f1; 2; . . . ; ng n fig. Obviously, Sn;k n Sin;k is connected. Since every vertex in Sin;k n F i has exactly one neighbor in Sn;k n Sin;k , and this neighbor is not faulty, Sn;k n F is connected, a contradiction. (3) There exists some subgraph, say Sin;k , has fi ¼ n 2 faulty vertices.
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Let u be the unique vertex in F n F i . Clearly, Sn;k n ðSin;k [ fugÞ, as explained in case (1) above, is connected. If Sin;k n F i is connected, Sn;k n F is connected, the reason is that the subgraph Sin;k n F i has at least two vertices, and at least one vertex is adjacent to one non-faulty vertex in Sn;k n ðSin;k [ fugÞ. If Sin;k n F i is not connected, we may, without loss of generality, assume that Sin;k n F i is divided into m disjoint connected components, say, C 1 ; C 2 ; . . . ; C m . If jC j j P 2 (1 6 j 6 m), C j has at least one vertex which is adjacent to one non-faulty vertex in Sn;k n ðSin;k [ fugÞ. If jC j j ¼ 1, we assume that C j ¼ fv g. If ðu; v Þ R EðSn;k Þ; v connects to the component Sn;k n ðSin;k [ fugÞ; otherwise, C j ¼ fv g is isolated. While the number of such C j is exactly one since we have jF n F i j ¼ 1. So we conclude that if Sn;k n F is disconnected, it has exactly two components, one of which has only one vertex, and its only disconnecting set is the set of the neighbors of this vertex. Hence, Sn;k is tightly ðn 1Þ-super connected. h Theorem 2. Let F be a faulty vertex set of Sn;k ðn P 5; k P 3; n k P 2Þ with jFj 6 n þ k 3. Then the survival graph Sn;k n F satisfies one of the following conditions: (1) Sn;k n F is connected; or (2) Sn;k n F has two components, one of which has exactly one vertex; or (3) Sn;k n F has two components, one of which is the spanning subgraph K 2 ½u; v with a 1-edge. Moreover, F ¼ NðuÞ [ Nðv Þ fu; v g.
Proof. Denote F i ¼ VðSin;k Þ \ F, and fi ¼ jF i j for 8i 2 f1; 2; . . . ; ng. We have
Pn
i¼1 fi
¼ jFj 6 n þ k 3, and discuss as follows.
(1) fi 6 n 3 for 8i 2 f1; 2; . . . ; ng. For 8i 2 f1; 2; . . . ; ng; Sin;k n F i is connected, since Sin;k is n-2-connected. The proof follows the idea of case (1) of Theorem 1. Since there are ðn 2Þ!=ðn kÞ! matching edges between two subgraphs Sin;k and Sjn;k for 8i – j 2 f1; 2; . . . ; ng, and 2ðn 2Þ!=ðn kÞ! P 2ðn 2Þ P n þ k 2 > n þ k 3. The subgraphs Sin;k and Sjn;k are connected by matching edges directly or indirectly through the help of some subgraphs Sxn;k and Syn;k with the matching edges between Sin;k and Sxn;k , the matching edges between Sxn;k and Syn;k , and the matching edges between Syn;k and Sjn;k , where x – y 2 f1; 2; . . . ; ng n fi; jg. Thus, Sn;k n F is connected. (2) There exists some subgraph, say Sin;k , which has fi P n 2 faulty vertices. There is at most one such subgraph, say Sin;k , otherwise there are at least 2n 4 > n þ k 3 faulty vertices, a contradiction. S The subgraph Sjn;k n F j stays connected as Sjn;k is ðn 2Þ-connected, but there are at most jF i j 6 j j–i F j j 6 n þ k 3 j ðn 2Þ ¼ k 1 6 n 3 faulty vertices in Sn;k for 8j 2 f1; 2; . . . ; ng n fig. Since there are ðn 2Þ!=ðn kÞ! matching edges between two subgraphs Sxn;k and Syn;k for x – y 2 f1; 2; . . . ; ng n fig, and ðn 2Þ!=ðn kÞ! P n 2 P k P k 1, Sxn;k and Syn;k are connected to each other directly. Thus, Sn;k n ðSin;k [ ðF n F i ÞÞ is connected. If fi ¼ n þ k 3, there is no faulty vertex outside Sin;k . Every vertex in Sin;k F i has one fault-free neighbor in Sn;k n ðSin;k [ ðF n F i ÞÞ, so Sn;k n F is connected. If fi ¼ n þ k 4, we may, without loss of generality, assume that F n F i ¼ fug and Sin;k n F i is divided into m disjoint connected components, say C 1 ; C 2 ; . . . ; C m . If jVðC j Þj P 2 (1 6 j 6 m), C j has at least one vertex adjacent to a fault-free vertex in Sn;k n ðSin;k [ fugÞ. So C j connects to the connected part Sn;k n ðSin;k [ fugÞ. If jVðC j Þj ¼ 1, we assume that C j ¼ fv g. Obviously, if ðu; v Þ R EðSn;k Þ; C j connects to the part Sn;k n ðSin;k [ fug); otherwise, C j ¼ fv g is isolated, but the number of such C j is exactly one, otherwise, n þ k 3 P n 2 þ n 2, a contradiction. Thus, if Sn;k n F is disconnected, it has exactly two components, one of which has only one vertex. Now, we assume that fi 6 n þ k 5. First, we suppose that Sin;k n F i is connected. Since Sin;k has ðn 1Þ!=ðn kÞ! vertices, each of which has exactly one cross edge connecting to Sn;k n Sin;k , and ðn 1Þ!=ðn kÞ! ðn þ k 3Þ > ðn 1Þðn 2Þ ðn þ k 3Þ > 0; Sn;k n F is connected. Now, we assume that Sn;k n F is disconnected. Except for the maximum component which has Sn;k n ðSin;k [ ðF n F i ÞÞ as its part, C 1 ; C 2 ; . . . ; C m are the disjoint connected components of Sn;k n F. Obviously, C j Sjn;k n F i ; j ¼ 1; 2 . . . ; m. We see jVðC j Þj 6 2 for j ¼ 1; 2; . . . ; m; otherwise, we suppose that jVðC j Þj P 3 for some j. Because the number of i-neighbor of u (8u 2 VðC j Þ) is k 1, and the number of 1-neighbor of u is n k. jF j j P jN Sj ðC j Þj P 3ðk 2Þ þ n k 2 ¼ n þ 2k 8. jVðC j Þj 6 n þ k 3 jF j j ¼ 5 k 6 2, a contradiction. Furthermore, n;k m ¼ 1; otherwise, if m P 2; n þ k 3 ¼ jFj P n 1 þ n 1 ¼ 2n 4, a contradiction. In detail, C 1 has one vertex or two vertices. We discuss as follows. If C 1 consists of one singleton u, apart from the maximum connected component, Sn;k n F has one isolated vertex u. If C 1 consists of two vertices u; v with one i-edge, n þ k 3 P jN Si ðC 1 Þj ¼ 2ðk 2Þ þ 2ðn kÞ ¼ 2n 4, a contradiction. n;k If C 1 consists of two vertices with one 1-edge. jN Si ðC 1 Þ [ N S nSi ðC 1 Þj ¼ 2ðk 2Þ þ n k 1 þ 2 ¼ n þ k 3, so we have n;k n;k n;k that Sn;k n F has two connected components, one of which is K 2 ½u; v with one 1-edge. h
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Theorem 3. Let F be a faulty vertex set of Sn;k ðn P 5; k P 3; n k P 2Þ with jFj 6 n þ 2k 6. Then the survival graph Sn;k n F satisfies one of the following conditions: (1) Sn;k n F is connected; or (2) Sn;k n F has two components, one of which has exactly one vertex or two vertices with one 1-edge; or (3) Sn;k n F has three components, two of which are singletons.
P Proof. Denote F i ¼ VðSin;k Þ \ F, and fi ¼ jF i j for 1 6 i 6 n. We have ni¼1 fi ¼ jFj 6 n þ 2k 6. We suppose that Sn;k n F is not connected, and try to prove that at least one of case (2) and (3) occurs:(1) 9i 2 f1; 2; . . . ; ng such that jF i j P n þ 2k 8. S Since jF j j 6 j j–i F j j ¼ jF F i j 6 n þ 2k 6 ðn þ 2k 8Þ 6 2, and 8j 2 f1; 2; . . . ; ng n fig; Sjn;k is ðn 2Þ-connected, so Sjn;k (n P 5) is connected. As there are ðn 2Þ!=ðn kÞ! matching edges between two subgraphs Sxn;k and Syn;k for 8x – y 2 f1; 2; . . . ; ng n fig, and ðn 2Þ!=ðn kÞ! P n 2 > 2 for n P 5; Sxn;k and Syn;k are directly connected. Thus, the subgraph Sn;k n ðSin;k [ FÞ is connected. We denote C the component that has Sn;k n ðSin;k [ FÞ as its part. S S If j j–i F j j ¼ 0; Sn;k n F is connected (a contradiction); if j j–i F j j ¼ 1, we suppose F n F i ¼ fug; Sin;k n F i has at most one vertex which is adjacent to u, so Sn;k n F is either connected (a contradiction); or has two components, one of which S is singleton; if j j–i F j j ¼ 2, we suppose that F n F i ¼ fu; v g. If Sin;k n F i is connected, Sn;k n F is connected (a contradiction); otherwise, at most two vertices of Sin;k n F i can be adjacent to u and v, since every vertex of Sin;k has exactly one neighbor outside Sin;k . Thus, consider the possibilities whether u or (and) v is (are) connected to C, and the possibilities whether u and v are adjacent, we conclude that the remained component consists of one singleton, two singletons or K 2 ½u; v with one 1-edge, except for the maximum component C.(2) 8i 2 f1; 2; . . . ; ng; jF i j 6 n þ 2k 9.(2.1) 8i 2 f1; 2; . . . ; ng; jF i j P n 3. The proof follows the idea of case (1) of Theorem 1. Since there are ðn 2Þ!=ðn kÞ! matching edges between two subgraphs Sin;k and Sjn;k for i – j 2 f1; 2; . . . ; ng, and 2ðn 2Þ!=ðn kÞ! P 2ðn 2Þ > 2ðn 3Þ, the subgraphs Sin;k and Sjn;k are connected by matching edges directly or indirectly through the help of some subgraphs Sxn;k and Syn;k with the matching edges between Sin;k and Sxn;k , the matching edges between Sxn;k and Syn;k , and the matching edges between Syn;k and Sjn;k , where x – y 2 f1; 2; . . . ; ng n fi; jg. Thus Sn;k n F is connected (a contradiction). (2.2) 9i 2 f1; 2; . . . ; ng; jF i j P n 2. (2.2.1) jF i j P n þ k 5; jF F i j 6 k 1 6 n 3. Obviously, Sn;k n ðSin;k [ FÞ is connected. our supposition is that: Sn;k n F is connected, except for the maximum component of Sn;k n F; C 1 ; C 2 ; . . . ; C m are the disjoint connected components of Sn;k n F. Obviously, C j Sin;k n F i ; j ¼ 1; 2; . . . ; m. First, we have jVðC j Þj 6 2 for j ¼ 1; 2; . . . ; m; otherwise, for 8b 2 VðC j Þ, the number of i-neighbors of b is k 1, and the number of 1neighbors of b is n k, jF j j P jN Sj ðC j Þj P 3ðk 2Þ þ n k 2 ¼ n þ 2k 8. jVðC j Þj 6 n þ 2k 6 jF j j ¼ n þ 2k 6 n;k ðn þ 2k 8Þ ¼ 2, a contradiction. So, we have jVðC j Þj 6 2. Furthermore, m 6 2; otherwise, if m P 3, we have n þ 2k 6 ¼ jFj P n 2 þ n 2 þ n 2 ¼ 3n 6, a contradiction. If D is a subset of VðSn;k Þ with at least 3 vertices, jNðDÞj P 3ðk 1Þ þ n k 2 ¼ n þ 2k 5 > n þ 2k 6. From the fact above, we can conclude that: if m ¼ 2; C 1 and C 2 each has exactly one vertex. If n > 2k 3; jNðC 1 [ C 2 Þj P 2n 2 1 > nþ 2k 6, a contradiction; otherwise apart from the maximum connected component, Sn;k n F has two isolated vertices. Now, we discuss the case of m ¼ 1, and C 1 has only one vertex or two vertices as follows: If C 1 consists of one singleton u, apart from the maximum connected component, Sn;k n F has one isolated vertex u; if C 1 consists of two vertices with one i-edge, jN Si ðC 1 Þj ¼ 2ðk 2Þ þ 2ðn kÞ ¼ 2n 4. If n 6 2k 2, 2n 4 6 n þ 2k 6; Sn;k n F n;k has one small component K 2 ½u; v with one 1-edge. If n > 2k 2; 2n 4 > n þ 2k 6, which deduces a contradiction; if C 1 consists of two vertices with one 1-edge. jN Si ðC 1 Þj ¼ 2ðk 1Þ þ n k 1 ¼ n þ k 3 6 n þ 2k 6, apart from n;k Sn;k n ðSin;k [ ðF n F i ÞÞ, Sn;k n F has one small component K 2 ½u; v with one 1-edge. (2.2.2) n 2 6 jF i j 6 n þ k 5 ¼ ðn 1Þ þ ðk 1Þ 3. Obviously, the number of such i’s is at most two; otherwise, jFj P 3n 6 > n þ 2k 6, a contradiction. We are confronted with two possibilities as follows: If there is only one i such that n 2 6 jF i j 6 n þ k 5, then by Theorem 2, Sin;k n F i is connected or disconnected, but in the latter case, Sin;k n F i has two connected components A1 ; A2 , and A1 consists of a singleton vertex, say u. Since jF j j 6 n 3 for 8j 2 f1; 2; . . . ; ng n fig; Sn;k n ðSin;k [ FÞ is connected. Denote by C the component that contains Sn;k n ðSin;k [ FÞ. Since ðn 1Þ!=ðn kÞ! 1 ðn þ 2k 6Þ > 1 for n P 5. A2 belongs to C. If the singleton u is connected to C, then Sn;k n F is connected, a contradiction. Thus, apart from C; Sn;k n F has one isolated vertex u. If there exist i and j such that n 2 6 jF i j 6 n þ k 5 and n 2 6 jF j j 6 n þ k 5, then by Theorem 2, Sin;k n F i is connected or disconnected, but in the latter case, Sin;k n F i has two connected components A1 ; A2 , and A1 consists of a singleton vertex, say u. Sjn;k n F j is connected or disconnected, but in the latter case, Sjn;k n F j has two connected components B1 ; B2 , and B1 consists of a singleton vertex, say v. Since jF l j 6 n 3 for 8l 2 f1; 2; . . . ; ng n fi; jg; Sn;k n ðSin;k [ Sjn;k [ FÞ is connected. Denote by C the component that contains Sn;k n ðSin;k [ Sjn;k [ FÞ. Since ðn 1Þ!=ðn kÞ! 1 ðn 2Þ ðn 4Þ > 1 for n P 5. A2 belongs to C. Likewise, B2 belongs to C. Thus, consider the possibilities whether u or (and) v is (are) connected to C, and the possibilities whether u and v are adjacent, one can conclude that the remained component consists of one singleton, two singletons or K 2 ½u; v with one 1-edge, except for the maximum component C. h
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Theorem 4. Let F VðSn;2 Þ ðn P 4Þ with jFj ¼ n 1. Then, the survival graph Sn;2 n F satisfies one of the following conditions: (1) Sn;2 n F is connected; or (2) Sn;2 n F has two components, the smaller one, say C; C 2 fK i j1 6 i 6 ng, where K i is the complete graph with order i. Proof. We suppose that the survival graph Sn;2 n F is not connected, and let F i ¼ VðSin;k Þ \ F, and fi ¼ jF i j for 1 6 i 6 n. ObviP ously, ni¼1 fi ¼ jFj ¼ n 1. We may, without loss of generality, assume that Sn;2 n F is divided into m disjoint connected components, say, C 1 ; C 2 ; . . . ; C m . Claim 1. For 8i 2 f1; 2; . . . ; mg; C i is in some Sjn;2 with j 2 f1; 2; . . . ; ng. if this is not true, we may suppose that C i ¼ C xi [ C yi , and C xi ¼ C i \ Sxn;2 ; C yi ¼ C i \ Syn;2 , where x – y 2 f1; 2; . . . ; ng. Since Sxn;2 is a complete graph, jVðC xi Þj ¼ jSxn;2 j fx ; similarly, jVðC yi Þj ¼ jSyn;2 j fy . As there is exactly one cross edge between Sxn;2 and Syn;2 , to separate C i from other part, it has
fx þ fy þ jVðC i Þj 2 ¼ n 1 jVðC xi Þj þ n 1 jVðC yi Þj þ jVðC i Þj 2 ¼ 2n 4 faulty vertices, where n P 4. Thus, we have 2n 4 > n 1 ¼ jFj, which is a contradiction. Similarly, the vertices of C i can not be distributed in more than two subgraphs. If C i ffi K q Sjn;2 , to separate C i from Sjn;2 , it has to get rid of jVðSjn;2 Þj jVðC i Þj ¼ n 1 q vertices. As every vertex of C i has exactly one cross edge connecting to Sn;2 n Sjn;2 it need to get rid of q vertices in NðC i Þ \ ðSn;2 n Sjn;2 Þ. So there are no surplus faulty vertices in F. This means that m ¼ 1, and C 1 ffi K q is the only connected component except for the largest connected component in Sn;2 n F. h Theorem 4 tells that Sn;2 is not tightly super connected, and its fault tolerance is inferior to Sn;k (n P 5; k P 3; n k P 2). In the next section, we only discuss the conditional diagnosability of Sn;k (n P 5; k P 3; n k P 2). 4. The conditional diagnosability of S n;k The comparison diagnosis strategy can be modeled as a multi-graph M ¼ ðV; CÞ, where V is the same node set defined as in G; C is the labelled edge set. A labelled edge ðu; v Þw is said to belong to C if (u, v) is an edge labeled by w, which implies that the processors u, v are compared by processor w. Since different comparators can compare the same pair of processors, M is a multi-graph. Denote the comparison result as rððu; v Þw Þ such that rððu; v Þw Þ ¼ 0 if the outputs of u and v agree, and rððu; v Þw Þ ¼ 1 if the outputs disagree. If the comparator w is fault-free and rððu; v Þw Þ ¼ 0, the processors u and v are fault-free; while rððu; v Þw Þ ¼ 1, at least one of the three processors u; v and w is faulty. The collection of the comparison results defined as a function r : C ! f0; 1g, is called the syndrome of the diagnosis. A subset F(V is said to be compatible with a syndrome r if r can arise from the circumstance that all vertices in F are faulty and all vertices in V n F are fault-free. A faulty comparator can lead to unreliable results, so a set of faulty vertices may produce different syndromes. Let rF ¼ frjr is compatible with F}. Two distinct subsets F 1 and F 2 of VðGÞ are said to be indistinguishable if and only if rF1 \ rF 1 – /; otherwise, both of F 1 and F 2 are said to be distinguishable. There are several different ways to verify whether a system is t-diagnosable under the comparison approach. The following lemma obtained by Sengupta and Dahbura [14] gives necessary and sufficient conditions to ensure distinguishability. Lemma 1. [14] Let G be a graph. For any two distinct subsets F 1 ; F 2 of VðGÞ. ðF 1 ; F 2 Þ is a distinguishable pair if and only if at least one of the following conditions is satisfied. (1) There are two distinct vertices u; w 2 VðGÞ ðF 1 [ F 2 Þ and there is a vertex v 2 F 1 DF 2 such that ðu; v Þw 2 C; (2) There are two distinct vertices u and v 2 F 1 n F 2 and there is a vertex w 2 VðGÞ ðF 1 [ F 2 Þ such that ðu; v Þw 2 C; or (3) There are two distinct vertices u; v 2 F 2 n F 1 and there is a vertex w 2 VðGÞ ðF 1 [ F 2 Þ such that ðu; v Þw 2 C. Lin et al. [15] introduced the so-called conditional diagnosability of a system under the situation that no set of faulty vertices can contain all neighbors of any vertex in the system. A faulty set F VðGÞ is called a conditional faulty set if N G ðv Þ Ü F for every vertex v 2 VðGÞ. A system GðV; EÞ is said to be conditionally t-diagnosable if F 1 and F 2 are distinguishable for each pair of distinct conditional faulty set F 1 and F 2 with jF 1 j 6 t; jF 2 j 6 t. The maximum value of t such that G is conditionally tdiagnosable is called the conditional diagnosability of G, denoted by t C ðGÞ. It is trivial that tC ðGÞ P tðGÞ. By the operation of 1-edges in Sn;k , there exist three vertices u; v ; w 2 VðSn;k Þ, such that ðu; v ; wÞ in a cycle of length 3, We set A ¼ N½u; v ; w; F 1 ¼ A fw; v g, and F 2 ¼ A fu; wg. We get jF 1 j ¼ jF 2 j ¼ 3ðk 1Þ þ n k 2 þ 1 ¼ n þ 2k 4, and jF 1 F 2 j ¼ jF 2 F 1 j ¼ 1. It is easy to check that F 1 and F 2 are two conditional faulty sets, and F 1 and F 2 are indistinguishable. Hence we have the following result: Theorem 5. tc ðSn;k Þ 6 n þ 2k 5 for n P 5; k P 3; n k P 2. The following two lemmas are necessary in our context. Lemma 2. Let F 1 and F 2 be any two distinct conditional fault-sets of Sn;k ðn P 5; k P 3; n k P 2Þ with jF 1 j 6 n þ 2k 5; jF 2 j 6 n þ 2k 5. Denote by H the maximum component of Sn;k F 1 \ F 2 . Then, for every vertex u 2 F 1 DF 2 ; u 2 H.
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Proof. Without loss of generality, we assume that u 2 F 1 F 2 . Since F 2 is a conditional faulty set, there is a vertex v 2 ðSn;k F 2 Þ fug such that ðu; v Þ 2 EðSn;k Þ. Suppose that u is not a vertex of H. Then v is not in H, so u and v are in one small component of Sn;k F 1 \ F 2 . Since F 1 and F 2 are distinct, we have jF 1 \ F 2 j 6 n þ 2k 6. Hence fu; v g forms a component K 2 in Sn;k F 1 \ F 2 by Theorem 3, i.e., the vertex u is the unique neighbor of v in Sn;k F 1 \ F 2 . This is a contradiction since F 1 is a conditional faulty set, but all the neighbors of v are faulty in Sn;k F 1 . h Lemma 3 [15]. Let G be a graph with dðGÞ P 2, and let F 1 and F 2 be any two distinct conditional faulty sets of G with F 2 F 1 . Then, ðF 1 ; F 2 Þ is a distinguishable conditional pair under the comparison diagnosis model. Lemma 4. Let F 1 and F 2 be any two distinct conditional faulty sets of Sn;k ðn P 5; k P 3; n k P 2Þ with jF 1 j 6 n þ 2k 5 and jF 2 j 6 n þ 2k 5. Then, ðF 1 ; F 2 Þ is a distinguishable conditional pair under the comparison diagnosis model. Proof. By Lemma 3, ðF 1 ; F 2 Þ is a distinguishable conditional pair of Sn;k if F 1 F 2 or F 2 F 1 . Thus, we assume that jF 1 F 2 j P 1, and jF 2 F 1 j P 1. Let S ¼ F 1 \ F 2 . Then we have jSj 6 n þ 2k 6. Let H be the largest connected component of Sn;k F 1 [ F 2 . By Lemma 2, every vertex in F 1 DF 2 is in H. We claim that H has a vertex u outside F 1 [ F 2 that has no neighbor in S. Since every vertex has degree n 1, the vertices in S have at most ðn 1ÞjSj neighbors in H. There are at most 2ðn þ 2k 6Þ jSj vertices in F 1 [ F 2 and at most two vertices of Sn;k S may not belong to H by Theorem 4. Since the fact is jSj 6 n þ 2k 6, we have
n! n! ð2ðn þ 2k 6Þ jSjÞ 2 P ðn 2ÞjSj 2ðn þ 2k 6Þ 2 ðn kÞ! ðn kÞ! n! P ðn þ 2k 6Þðn 2Þ 2ðn þ 2k 6Þ 2 P 4 for n P 5; k ðn kÞ! P 3; n k P 2: Thus, there must be some vertex of H outside F 1 [ F 2 , which has no neighbors in S. Let u be such a vertex. If u has no neighbor in F 1 [ F 2 , then we can find a path of length at least two within H to a vertex v in F 1 [ F 2 . We may assume that v is the first vertex of F 1 DF 2 on this path, and let q and w be the two vertices on this path immediately before v (we may have u ¼ q), so q and w are not in F 1 [ F 2 . The existence of the edges ðq; wÞ and ðw; v Þ shows that ðF 1 ; F 2 Þ is a distinguishable conditional pair of Sn;k by Lemma 1. Now we assume that u has a neighbor in F 1 DF 2 . Since the degree of u is at least 3, and u has no neighbor in S, there are three possibilities: (1) u has two neighbors in F 1 F 2 ; or (2) u has two neighbors in F 2 F 1 ; or (3) u has at least one neighbor outside F 2 [ F 1 . In each subcase above, Lemma 1 implies that ðF 1 ; F 2 Þ is a distinguishable conditional pair of Sn;k under the comparison diagnosis model, finishing the proof. h Theorem 6. tc ðSn;k Þ ¼ n þ 2k 5 for n P 5, k P 3; n k P 2. The class of alternating group networks was introduced in the late 1990’s as an alternative to the alternating group graphs as interconnection networks. Cheng et al. [30] showed that this topology is in fact isomorphic to the ðn; n 2Þ-star graph, a member of the well-known (n, k)-star graphs. Zhou and Xiao [31] derived that the conditional diagnosability of alternating group networks AN n under the comparison model. In fact, their result is a special case of theorem 6. Theorem 7 [31]. t c ðANn Þ ¼ 3n 9 for n P 5. 5. Conclusion The paper is interested in the study of fault resiliency of interconnection networks. The fault resiliency can be used to reveal the conditional connectivity of high order, and also to evaluate the conditional diagnosability under the comparison model. This method can be applied to other complex network structures. Acknowledgments The author thanks the anonymous referees for their valuable suggestions. This research was financially supported by National Natural Science Funds of China (No. 61072080) and the Key Project of Fujian Provincial Universities services to the western coast of the straits-Information Technology Research Based on Mathematics.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The conditional fault diagnosability of (n, k)-star graphs Shuming Zhou Key Laboratory of Network Security and Cryptology, Fujian Normal University, Fuzhou, Fujian 350007, PR China College of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350007, PR China
a r t i c l e
i n f o
Keywords: Fault tolerance Comparison diagnosis Conditional diagnosability (n, k)-Star graphs
a b s t r a c t The growing size of the multiprocessor system increases its vulnerability to component failures. It is crucial to locate and replace the faulty processors to maintain a system’s high reliability. The fault diagnosis is the process of identifying faulty processors in a system through testing. This paper shows that the largest connected component of the survival graph contains almost all the remaining vertices when a lot of faulty vertices occur in the (n, k)-star graph Sn;k ðn P 5; k P 3; n k P 2Þ. Based on this fault resiliency, it establishes the conditional fault diagnosability of Sn;k under the comparison model. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Distributed processor architectures offer the potential advantage of high speed, provided that they are highly fault tolerant and reliable, and have good communication between remote processors. An important component of such a distributed system is its graph topology, which defines the inter-processor communication architecture. Fault tolerance can be especially important for interconnection networks, since computers may fail, creating faults in the network. To be reliable, the rest of the network should stay connected. Obviously, this can only be guaranteed if the number of faults is smaller than the smallest degree in the network. When the number of faults is larger than the smallest degree, some extensions of connectivity are necessary, since the graph may become disconnected. Thus, several types of connectivity were introduced and examined for various classes of graphs, such as loosely super connectedness and tightly super connectedness [1], super connectivity [2], super k-restricted edge connectivity [3]. As we increase the number of faults in the graph, it is desirable that most of the network stays connected, with at most a few processors separated from the rest, since then the network will continue to be able to function. Many interconnection networks have been examined in this aspect, when the number of faults is about twice the smallest degree [4,5]. One can even go further and ask what happens when more vertices are deleted. This was examined for the hypercube in [6–8] and for certain Cayley graphs generated by transpositions in [9], and it was shown that the resulting network will have a large component containing almost all vertices. The process of identifying faulty processors in a system by analyzing the outcomes of available inter-processor tests is called system-level diagnosis. The foundation of system diagnosis and an original diagnostic model, namely the PMC model, were established in a classic paper by Preparata et al. [10]. Its target is to identify the exact set of all faulty nodes before their repair or replacement. All tests are performed between two adjacent processors, and it was assumed that a test result is reliable (respectively, unreliable) if the processor that initiates the test is fault-free (respectively, faulty). The comparison-based diagnosis models, first proposed by Malek [11] and Chwa and Hakimi [12], have been considered to be a practical approach for fault diagnosis in multiprocessor systems. In these models, the same job is assigned to a pair of processors in the system and their outputs are compared by a central observer. This central observer performs diagnosis using the outcomes of these
⇑ Address: College of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350007, PR China. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.021
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comparisons. Maeng and Malek [13] extended Malek’s comparison approach to allow the comparisons carried out by the processors themselves. Sengupta and Dahbura [14] developed this comparison approach such that the comparisons have no central unit involved. Lin et al. [15] introduced the conditional diagnosis. By evaluating the size of connected components, they obtained that the conditional diagnosability of Star graph Sn is 3n 7 which is about three times larger than the classical diagnosability of star graphs. Using the evaluation of the size of connected components, Hsu et al. [16] have recently proved that the conditional diagnosability of hypercube is 3n 5. This idea was attributed to Lai et al. [17] who is the first to use a restricted diagnosis strategy. They obtained that the conditional diagnosability of the hypercubes Q n is 4n 7 under the PMC model. Furthermore, Hsu et al. [16] exposed the difference between these two conditional diagnosis models. This paper establishes the conditional diagnosability of Sn;k (n P 5; k P 3; n k P 2) under the comparison diagnosis model based on the fault tolerance of (n, k)-star graph Sn;k . The rest of this paper is organized as follows. Section 2 introduces some definitions, notations and the structure of (n, k)-star graph Sn;k . Section 3 is devoted to the fault resiliency of Sn;k ; and Section 4 concentrates on the conditional diagnosability of Sn;k . Section 5 concludes the paper. 2. Preliminaries Throughout this paper, we use a graph G ¼ GðV; EÞ to represent an interconnection network, where each node u 2 V denotes a processor and each edge ðu; v Þ 2 E denotes a link between nodes u and v. Let S be a subset of VðGÞ. The subgraph of G induced by S, denoted by G½S, is the graph with the vertex set S \ VðGÞ and the edge set fðu; v Þjðu; v Þ 2 EðGÞ; u; v 2 Sg. For any subset F V, the notation G n F (or G F) represents the graph obtained by removing the vertices in F from G and deleting those edges with at least one end vertex in F simultaneously. If G n F is disconnected, F is called a vertex cut or a separating set. The components of G n F are its maximal connected subgraphs. For any node u of G; NðuÞ denotes the set of all its neighS boring nodes, i.e., NðuÞ ¼ fv jðu; v Þ 2 Eg. For any set F V, let NðFÞ ¼ u2F NðuÞ F; N½F ¼ NðFÞ [ F. For brevity, N½u ¼ NðuÞ [ fug, Nðfu; v gÞ and N½fu; v g are written as Nðu; v Þ and N½u; v . The symmetric difference of any two sets F 1 and F 2 is defined as the set F 1 DF 2 ¼ ðF 1 F 2 Þ [ ðF 2 F 1 Þ. Network reliability is one of the major factors in designing the topology of an interconnection network. The hypercube and its variants were the first major class of interconnection networks. The n-star graph (Sn for short), which proposed by Akers and Krishnamurthy is an attractive alternative to the hypercube [18]. An Sn is a Cayley graph with a regular and hierarchical structure; for a similar number of vertices, the graph has a lower vertex degree, a smaller diameter, and a shorter average distance than the comparable hypercube [19]. However, with the restriction on the number of vertices n!, there is a large gap between n! and (n + 1)! for expanding an Sn to an Snþ1 . To relax the restriction of the numbers of vertices n! in Sn , a generalized version of the star graph, the (n, k)-star graph, was proposed by Chiang and Chen [20]. An Sn;k preserves many attractive properties of Sn such as vertex symmetry, hierarchical structure, maximal fault tolerance, and simple shortest routing. The two parameters n and k can be tuned to make a suitable choice for the number of vertices in the network and for the degree/diameter tradeoff. An n! Sn;k is regular of degree n 1, and the number of vertices is ðnkÞ! . They form a hierarchical family of graphs, each of which is node-symmetric; they can be recursively decomposed in a number of ways; they have a simple shortest-path routing algorithm [20]; the node-connectivity of Sn;k is n 1 [21]; there is an exact formula for their diameters, and their fault-diameters are at most their fault-free diameters plus 3 [21]; Hsu et al. [22] proved that Sn;k F is Hamiltonian (resp. Hamiltonian connected) if jFj 6 n 3 (resp. jFj 6 n 4) with F VðSn;k Þ [ EðSn;k Þ and n k P 2; Lin and Duh [23] also proved that the (n, k)-star graph Sn;k is super spanning connected if n > 3 and n k P 2. Chen et al. [24] successfully gave a method for constructing vertex-disjoint paths in (n, k)-star graphs. Chen et al. [24] showed that (n, k)-star graph Sn;k is 6-weak-vertex-pancyclic. Recently, Xiang and Stewart [25] presented an algorithm to find one-to-many node-disjoint paths in (n, k)-star graphs. Now, we formally present the structure of the (n, k)-star graph Sn;k . For simplicity, we use hni to denote the set f1; 2; . . . ; ng. The (n, k)-star graph, denoted by Sn;k , is a graph with the vertex set VðSn;k Þ ¼ fu1 u2 uk1 uk jui 2 hni, and ui – uj for i – jg, and adjacency is defined as follows: A vertex u1 u2 uk1 uk is adjacent to. (1) the vertex ui u2 ui1 u1 uiþ1 uk , where 2 6 i 6 k; (2) the vertex xu2 uk1 uk , where x 2 hni fu1 u2 uk1 uk g. The edges of type (1) are referred to as i-edges and the two end vertices are i-neighbors to each other. The edges of type (2) are referred to as 1-edges and, the two end vertices are 1-neighbors to each other [19]. The structure of an S4;2 is shown in Fig. 1. An Sn;k can be formed by interconnecting n Sn1;k1 ’s, that is, an Sn;k can be decomposed into Sn1;k1 ’s along any dimension i, and it can also be decomposed into n vertex disjoint Sn1;k1 ’s in k 1 different ways by fixing one symbol in any position i; 2 6 i 6 k. We denote Sin;k the subgraph which fixes the symbol i in the last position k. Obviously, Sin;k ffi Sn1;k1 . This decomposition can be recursively carried out on each Sin;k to obtain smaller subgraphs. The graph Sin;k is undirected and vertex-symmetric, but two different types of edges (1-edge and i-edge with 2 6 i 6 k) in the graph Sn;k prevent being edge symmetric. 3. Fault tolerance of (n, k)-star graph S n;k The connectivity jðGÞ of a graph G ¼ GðV; EÞ is the minimum number of nodes whose removal results in a disconnected or a trivial (one node) graph. The connectivity jðGÞ of G is an important parameter to measure the fault tolerance of the
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Fig. 1. S4,2.
network, while it has an obvious deficiency in that it tacitly assume that all elements in any subset of G can potentially fail at the same time. To compensate for this shortcoming, it would seem natural to generalize the classical connectivity by introducing some conditions or restrictions on the separating set S and/or the components of G n S. A k-regular graph is maximally connected if it is k-connected. A k-regular graph is (loosely) super k-connected if any one of its minimum separating sets is a set of the neighbors of some vertex. If, in addition, the deletion of a minimum separating set results in a graph with two components (one of which has only one vertex), then the graph is tightly super k-connected. Note that a k-regular graph can be loosely super connected but not tightly super connected [1]. Consequently, Esfahanian introduced the concepts of the restricted cut and the restricted connectivity of a graph [26]. A restricted vertex set S is a restricted vertex-cut if G n S is disconnected, and no component is an isolated vertex. The restricted vertex connectivity of a graph G, denoted by j0 ðGÞ, is the minimum cardinality of a restricted vertex-cut. It has been shown that if a network possesses the restricted connectivity property, it is more reliable and has the lower vertex failure comparing to that has only the super connectivity property. Usually, if the surviving graph contains a large connected component, it may be used as the functional subsystem, without incurring severe performance degradation. Thus, in evaluating a distributed system, it is indispensable to estimate the size of the maximal connected components of the underlying graph when the structure begins to lose processors. Yang et al. [6–8] proved that the hypercube Q n with a set F of at most 2n 3 failing processors has a component of size 2n jFj 1; and that it has a component of size 2n jFj 2 if jFj 6 3n 6. Similar result of Star graph Sn was also obtained by Yang et al. [27,28]. Cheng et al., [29,30] gave a more detail on this result as follows: Let F be a set of vertices of at most 2n 4, if Sn F is not connected, Sn F has two components, one of which has exactly one vertex, or has two vertices with a edge, and in the latter case, F is the union of the neighbor sets of these two vertices excluding themselves. Cheng and Lipták [9] generalized this result for star graph with linearly many faults. In this section, we discuss the fault resilience of Sn;k . Theorem 1. The (n, k)-star graph Sn;k ðn P 5; k P 3; n k P 2Þ is tightly (n 1)-super connected.
Proof. As the connectivity of the (n, k)-star graph Sn;k (n P 5; k P 3; n k P 2) is n 1, we suppose that F is a subset of P VðSn;k Þ with jFj ¼ n 1, and Sn;k n F is not connected. Let F i ¼ VðSin;k Þ \ F, and fi ¼ jF i j for 1 6 i 6 n. So ni¼1 fi ¼ jFj ¼ n 1. We discuss as follows: (1) fi 6 n 3 for 8i 2 f1; 2; . . . ; ng. For 8i 2 f1; 2; . . . ; ng; Sin;k n F i is connected since Sin;k is n 2-connected. If Sin;k n F i is connected to Sjn;k n F j for 0 8i – j 2 f1; 2; . . . ; ng; Sn;k n F is connected; otherwise, there exist i0 ; j0 2 f1; 2; . . . ; ng such that Sin;k n F i0 is not directly j0 i ðn2Þ! connected to Sn;k n F j0 . As there are ðnkÞ! matching edges between two subgraphs Sn;k and Sjn;k for 8i – j 2 f1; 2; . . . ; ng, and ðn2Þ! P n 2 P 3 for n P 5, k P 3; n k P 2, there are at least n 2 faulty vertices which destroy ðnkÞ! j0 0 the matching edges between Sin;k n F i0 and Sn;k n F j0 , while there is at most one faulty vertex which can destroy another j0 i0 matching edge, so Sn;k n F i0 can connect to Sn;k n F j0 indirectly through the help of some subgraph (s) of fS1n;k ; j0 0 S2n;k ; . . . ; Snn;k g n fSin;k ; Sn;k g. Thus Sn;k n F is connected, a contradiction. (2) There exists some subgraph, say Sin;k , which has fi ¼ n 1 faulty vertices, consequently, F j ¼ VðSjn;k Þ \ F ¼ / for 8j 2 f1; 2; . . . ; ng n fig. Obviously, Sn;k n Sin;k is connected. Since every vertex in Sin;k n F i has exactly one neighbor in Sn;k n Sin;k , and this neighbor is not faulty, Sn;k n F is connected, a contradiction. (3) There exists some subgraph, say Sin;k , has fi ¼ n 2 faulty vertices.
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Let u be the unique vertex in F n F i . Clearly, Sn;k n ðSin;k [ fugÞ, as explained in case (1) above, is connected. If Sin;k n F i is connected, Sn;k n F is connected, the reason is that the subgraph Sin;k n F i has at least two vertices, and at least one vertex is adjacent to one non-faulty vertex in Sn;k n ðSin;k [ fugÞ. If Sin;k n F i is not connected, we may, without loss of generality, assume that Sin;k n F i is divided into m disjoint connected components, say, C 1 ; C 2 ; . . . ; C m . If jC j j P 2 (1 6 j 6 m), C j has at least one vertex which is adjacent to one non-faulty vertex in Sn;k n ðSin;k [ fugÞ. If jC j j ¼ 1, we assume that C j ¼ fv g. If ðu; v Þ R EðSn;k Þ; v connects to the component Sn;k n ðSin;k [ fugÞ; otherwise, C j ¼ fv g is isolated. While the number of such C j is exactly one since we have jF n F i j ¼ 1. So we conclude that if Sn;k n F is disconnected, it has exactly two components, one of which has only one vertex, and its only disconnecting set is the set of the neighbors of this vertex. Hence, Sn;k is tightly ðn 1Þ-super connected. h Theorem 2. Let F be a faulty vertex set of Sn;k ðn P 5; k P 3; n k P 2Þ with jFj 6 n þ k 3. Then the survival graph Sn;k n F satisfies one of the following conditions: (1) Sn;k n F is connected; or (2) Sn;k n F has two components, one of which has exactly one vertex; or (3) Sn;k n F has two components, one of which is the spanning subgraph K 2 ½u; v with a 1-edge. Moreover, F ¼ NðuÞ [ Nðv Þ fu; v g.
Proof. Denote F i ¼ VðSin;k Þ \ F, and fi ¼ jF i j for 8i 2 f1; 2; . . . ; ng. We have
Pn
i¼1 fi
¼ jFj 6 n þ k 3, and discuss as follows.
(1) fi 6 n 3 for 8i 2 f1; 2; . . . ; ng. For 8i 2 f1; 2; . . . ; ng; Sin;k n F i is connected, since Sin;k is n-2-connected. The proof follows the idea of case (1) of Theorem 1. Since there are ðn 2Þ!=ðn kÞ! matching edges between two subgraphs Sin;k and Sjn;k for 8i – j 2 f1; 2; . . . ; ng, and 2ðn 2Þ!=ðn kÞ! P 2ðn 2Þ P n þ k 2 > n þ k 3. The subgraphs Sin;k and Sjn;k are connected by matching edges directly or indirectly through the help of some subgraphs Sxn;k and Syn;k with the matching edges between Sin;k and Sxn;k , the matching edges between Sxn;k and Syn;k , and the matching edges between Syn;k and Sjn;k , where x – y 2 f1; 2; . . . ; ng n fi; jg. Thus, Sn;k n F is connected. (2) There exists some subgraph, say Sin;k , which has fi P n 2 faulty vertices. There is at most one such subgraph, say Sin;k , otherwise there are at least 2n 4 > n þ k 3 faulty vertices, a contradiction. S The subgraph Sjn;k n F j stays connected as Sjn;k is ðn 2Þ-connected, but there are at most jF i j 6 j j–i F j j 6 n þ k 3 j ðn 2Þ ¼ k 1 6 n 3 faulty vertices in Sn;k for 8j 2 f1; 2; . . . ; ng n fig. Since there are ðn 2Þ!=ðn kÞ! matching edges between two subgraphs Sxn;k and Syn;k for x – y 2 f1; 2; . . . ; ng n fig, and ðn 2Þ!=ðn kÞ! P n 2 P k P k 1, Sxn;k and Syn;k are connected to each other directly. Thus, Sn;k n ðSin;k [ ðF n F i ÞÞ is connected. If fi ¼ n þ k 3, there is no faulty vertex outside Sin;k . Every vertex in Sin;k F i has one fault-free neighbor in Sn;k n ðSin;k [ ðF n F i ÞÞ, so Sn;k n F is connected. If fi ¼ n þ k 4, we may, without loss of generality, assume that F n F i ¼ fug and Sin;k n F i is divided into m disjoint connected components, say C 1 ; C 2 ; . . . ; C m . If jVðC j Þj P 2 (1 6 j 6 m), C j has at least one vertex adjacent to a fault-free vertex in Sn;k n ðSin;k [ fugÞ. So C j connects to the connected part Sn;k n ðSin;k [ fugÞ. If jVðC j Þj ¼ 1, we assume that C j ¼ fv g. Obviously, if ðu; v Þ R EðSn;k Þ; C j connects to the part Sn;k n ðSin;k [ fug); otherwise, C j ¼ fv g is isolated, but the number of such C j is exactly one, otherwise, n þ k 3 P n 2 þ n 2, a contradiction. Thus, if Sn;k n F is disconnected, it has exactly two components, one of which has only one vertex. Now, we assume that fi 6 n þ k 5. First, we suppose that Sin;k n F i is connected. Since Sin;k has ðn 1Þ!=ðn kÞ! vertices, each of which has exactly one cross edge connecting to Sn;k n Sin;k , and ðn 1Þ!=ðn kÞ! ðn þ k 3Þ > ðn 1Þðn 2Þ ðn þ k 3Þ > 0; Sn;k n F is connected. Now, we assume that Sn;k n F is disconnected. Except for the maximum component which has Sn;k n ðSin;k [ ðF n F i ÞÞ as its part, C 1 ; C 2 ; . . . ; C m are the disjoint connected components of Sn;k n F. Obviously, C j Sjn;k n F i ; j ¼ 1; 2 . . . ; m. We see jVðC j Þj 6 2 for j ¼ 1; 2; . . . ; m; otherwise, we suppose that jVðC j Þj P 3 for some j. Because the number of i-neighbor of u (8u 2 VðC j Þ) is k 1, and the number of 1-neighbor of u is n k. jF j j P jN Sj ðC j Þj P 3ðk 2Þ þ n k 2 ¼ n þ 2k 8. jVðC j Þj 6 n þ k 3 jF j j ¼ 5 k 6 2, a contradiction. Furthermore, n;k m ¼ 1; otherwise, if m P 2; n þ k 3 ¼ jFj P n 1 þ n 1 ¼ 2n 4, a contradiction. In detail, C 1 has one vertex or two vertices. We discuss as follows. If C 1 consists of one singleton u, apart from the maximum connected component, Sn;k n F has one isolated vertex u. If C 1 consists of two vertices u; v with one i-edge, n þ k 3 P jN Si ðC 1 Þj ¼ 2ðk 2Þ þ 2ðn kÞ ¼ 2n 4, a contradiction. n;k If C 1 consists of two vertices with one 1-edge. jN Si ðC 1 Þ [ N S nSi ðC 1 Þj ¼ 2ðk 2Þ þ n k 1 þ 2 ¼ n þ k 3, so we have n;k n;k n;k that Sn;k n F has two connected components, one of which is K 2 ½u; v with one 1-edge. h
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Theorem 3. Let F be a faulty vertex set of Sn;k ðn P 5; k P 3; n k P 2Þ with jFj 6 n þ 2k 6. Then the survival graph Sn;k n F satisfies one of the following conditions: (1) Sn;k n F is connected; or (2) Sn;k n F has two components, one of which has exactly one vertex or two vertices with one 1-edge; or (3) Sn;k n F has three components, two of which are singletons.
P Proof. Denote F i ¼ VðSin;k Þ \ F, and fi ¼ jF i j for 1 6 i 6 n. We have ni¼1 fi ¼ jFj 6 n þ 2k 6. We suppose that Sn;k n F is not connected, and try to prove that at least one of case (2) and (3) occurs:(1) 9i 2 f1; 2; . . . ; ng such that jF i j P n þ 2k 8. S Since jF j j 6 j j–i F j j ¼ jF F i j 6 n þ 2k 6 ðn þ 2k 8Þ 6 2, and 8j 2 f1; 2; . . . ; ng n fig; Sjn;k is ðn 2Þ-connected, so Sjn;k (n P 5) is connected. As there are ðn 2Þ!=ðn kÞ! matching edges between two subgraphs Sxn;k and Syn;k for 8x – y 2 f1; 2; . . . ; ng n fig, and ðn 2Þ!=ðn kÞ! P n 2 > 2 for n P 5; Sxn;k and Syn;k are directly connected. Thus, the subgraph Sn;k n ðSin;k [ FÞ is connected. We denote C the component that has Sn;k n ðSin;k [ FÞ as its part. S S If j j–i F j j ¼ 0; Sn;k n F is connected (a contradiction); if j j–i F j j ¼ 1, we suppose F n F i ¼ fug; Sin;k n F i has at most one vertex which is adjacent to u, so Sn;k n F is either connected (a contradiction); or has two components, one of which S is singleton; if j j–i F j j ¼ 2, we suppose that F n F i ¼ fu; v g. If Sin;k n F i is connected, Sn;k n F is connected (a contradiction); otherwise, at most two vertices of Sin;k n F i can be adjacent to u and v, since every vertex of Sin;k has exactly one neighbor outside Sin;k . Thus, consider the possibilities whether u or (and) v is (are) connected to C, and the possibilities whether u and v are adjacent, we conclude that the remained component consists of one singleton, two singletons or K 2 ½u; v with one 1-edge, except for the maximum component C.(2) 8i 2 f1; 2; . . . ; ng; jF i j 6 n þ 2k 9.(2.1) 8i 2 f1; 2; . . . ; ng; jF i j P n 3. The proof follows the idea of case (1) of Theorem 1. Since there are ðn 2Þ!=ðn kÞ! matching edges between two subgraphs Sin;k and Sjn;k for i – j 2 f1; 2; . . . ; ng, and 2ðn 2Þ!=ðn kÞ! P 2ðn 2Þ > 2ðn 3Þ, the subgraphs Sin;k and Sjn;k are connected by matching edges directly or indirectly through the help of some subgraphs Sxn;k and Syn;k with the matching edges between Sin;k and Sxn;k , the matching edges between Sxn;k and Syn;k , and the matching edges between Syn;k and Sjn;k , where x – y 2 f1; 2; . . . ; ng n fi; jg. Thus Sn;k n F is connected (a contradiction). (2.2) 9i 2 f1; 2; . . . ; ng; jF i j P n 2. (2.2.1) jF i j P n þ k 5; jF F i j 6 k 1 6 n 3. Obviously, Sn;k n ðSin;k [ FÞ is connected. our supposition is that: Sn;k n F is connected, except for the maximum component of Sn;k n F; C 1 ; C 2 ; . . . ; C m are the disjoint connected components of Sn;k n F. Obviously, C j Sin;k n F i ; j ¼ 1; 2; . . . ; m. First, we have jVðC j Þj 6 2 for j ¼ 1; 2; . . . ; m; otherwise, for 8b 2 VðC j Þ, the number of i-neighbors of b is k 1, and the number of 1neighbors of b is n k, jF j j P jN Sj ðC j Þj P 3ðk 2Þ þ n k 2 ¼ n þ 2k 8. jVðC j Þj 6 n þ 2k 6 jF j j ¼ n þ 2k 6 n;k ðn þ 2k 8Þ ¼ 2, a contradiction. So, we have jVðC j Þj 6 2. Furthermore, m 6 2; otherwise, if m P 3, we have n þ 2k 6 ¼ jFj P n 2 þ n 2 þ n 2 ¼ 3n 6, a contradiction. If D is a subset of VðSn;k Þ with at least 3 vertices, jNðDÞj P 3ðk 1Þ þ n k 2 ¼ n þ 2k 5 > n þ 2k 6. From the fact above, we can conclude that: if m ¼ 2; C 1 and C 2 each has exactly one vertex. If n > 2k 3; jNðC 1 [ C 2 Þj P 2n 2 1 > nþ 2k 6, a contradiction; otherwise apart from the maximum connected component, Sn;k n F has two isolated vertices. Now, we discuss the case of m ¼ 1, and C 1 has only one vertex or two vertices as follows: If C 1 consists of one singleton u, apart from the maximum connected component, Sn;k n F has one isolated vertex u; if C 1 consists of two vertices with one i-edge, jN Si ðC 1 Þj ¼ 2ðk 2Þ þ 2ðn kÞ ¼ 2n 4. If n 6 2k 2, 2n 4 6 n þ 2k 6; Sn;k n F n;k has one small component K 2 ½u; v with one 1-edge. If n > 2k 2; 2n 4 > n þ 2k 6, which deduces a contradiction; if C 1 consists of two vertices with one 1-edge. jN Si ðC 1 Þj ¼ 2ðk 1Þ þ n k 1 ¼ n þ k 3 6 n þ 2k 6, apart from n;k Sn;k n ðSin;k [ ðF n F i ÞÞ, Sn;k n F has one small component K 2 ½u; v with one 1-edge. (2.2.2) n 2 6 jF i j 6 n þ k 5 ¼ ðn 1Þ þ ðk 1Þ 3. Obviously, the number of such i’s is at most two; otherwise, jFj P 3n 6 > n þ 2k 6, a contradiction. We are confronted with two possibilities as follows: If there is only one i such that n 2 6 jF i j 6 n þ k 5, then by Theorem 2, Sin;k n F i is connected or disconnected, but in the latter case, Sin;k n F i has two connected components A1 ; A2 , and A1 consists of a singleton vertex, say u. Since jF j j 6 n 3 for 8j 2 f1; 2; . . . ; ng n fig; Sn;k n ðSin;k [ FÞ is connected. Denote by C the component that contains Sn;k n ðSin;k [ FÞ. Since ðn 1Þ!=ðn kÞ! 1 ðn þ 2k 6Þ > 1 for n P 5. A2 belongs to C. If the singleton u is connected to C, then Sn;k n F is connected, a contradiction. Thus, apart from C; Sn;k n F has one isolated vertex u. If there exist i and j such that n 2 6 jF i j 6 n þ k 5 and n 2 6 jF j j 6 n þ k 5, then by Theorem 2, Sin;k n F i is connected or disconnected, but in the latter case, Sin;k n F i has two connected components A1 ; A2 , and A1 consists of a singleton vertex, say u. Sjn;k n F j is connected or disconnected, but in the latter case, Sjn;k n F j has two connected components B1 ; B2 , and B1 consists of a singleton vertex, say v. Since jF l j 6 n 3 for 8l 2 f1; 2; . . . ; ng n fi; jg; Sn;k n ðSin;k [ Sjn;k [ FÞ is connected. Denote by C the component that contains Sn;k n ðSin;k [ Sjn;k [ FÞ. Since ðn 1Þ!=ðn kÞ! 1 ðn 2Þ ðn 4Þ > 1 for n P 5. A2 belongs to C. Likewise, B2 belongs to C. Thus, consider the possibilities whether u or (and) v is (are) connected to C, and the possibilities whether u and v are adjacent, one can conclude that the remained component consists of one singleton, two singletons or K 2 ½u; v with one 1-edge, except for the maximum component C. h
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Theorem 4. Let F VðSn;2 Þ ðn P 4Þ with jFj ¼ n 1. Then, the survival graph Sn;2 n F satisfies one of the following conditions: (1) Sn;2 n F is connected; or (2) Sn;2 n F has two components, the smaller one, say C; C 2 fK i j1 6 i 6 ng, where K i is the complete graph with order i. Proof. We suppose that the survival graph Sn;2 n F is not connected, and let F i ¼ VðSin;k Þ \ F, and fi ¼ jF i j for 1 6 i 6 n. ObviP ously, ni¼1 fi ¼ jFj ¼ n 1. We may, without loss of generality, assume that Sn;2 n F is divided into m disjoint connected components, say, C 1 ; C 2 ; . . . ; C m . Claim 1. For 8i 2 f1; 2; . . . ; mg; C i is in some Sjn;2 with j 2 f1; 2; . . . ; ng. if this is not true, we may suppose that C i ¼ C xi [ C yi , and C xi ¼ C i \ Sxn;2 ; C yi ¼ C i \ Syn;2 , where x – y 2 f1; 2; . . . ; ng. Since Sxn;2 is a complete graph, jVðC xi Þj ¼ jSxn;2 j fx ; similarly, jVðC yi Þj ¼ jSyn;2 j fy . As there is exactly one cross edge between Sxn;2 and Syn;2 , to separate C i from other part, it has
fx þ fy þ jVðC i Þj 2 ¼ n 1 jVðC xi Þj þ n 1 jVðC yi Þj þ jVðC i Þj 2 ¼ 2n 4 faulty vertices, where n P 4. Thus, we have 2n 4 > n 1 ¼ jFj, which is a contradiction. Similarly, the vertices of C i can not be distributed in more than two subgraphs. If C i ffi K q Sjn;2 , to separate C i from Sjn;2 , it has to get rid of jVðSjn;2 Þj jVðC i Þj ¼ n 1 q vertices. As every vertex of C i has exactly one cross edge connecting to Sn;2 n Sjn;2 it need to get rid of q vertices in NðC i Þ \ ðSn;2 n Sjn;2 Þ. So there are no surplus faulty vertices in F. This means that m ¼ 1, and C 1 ffi K q is the only connected component except for the largest connected component in Sn;2 n F. h Theorem 4 tells that Sn;2 is not tightly super connected, and its fault tolerance is inferior to Sn;k (n P 5; k P 3; n k P 2). In the next section, we only discuss the conditional diagnosability of Sn;k (n P 5; k P 3; n k P 2). 4. The conditional diagnosability of S n;k The comparison diagnosis strategy can be modeled as a multi-graph M ¼ ðV; CÞ, where V is the same node set defined as in G; C is the labelled edge set. A labelled edge ðu; v Þw is said to belong to C if (u, v) is an edge labeled by w, which implies that the processors u, v are compared by processor w. Since different comparators can compare the same pair of processors, M is a multi-graph. Denote the comparison result as rððu; v Þw Þ such that rððu; v Þw Þ ¼ 0 if the outputs of u and v agree, and rððu; v Þw Þ ¼ 1 if the outputs disagree. If the comparator w is fault-free and rððu; v Þw Þ ¼ 0, the processors u and v are fault-free; while rððu; v Þw Þ ¼ 1, at least one of the three processors u; v and w is faulty. The collection of the comparison results defined as a function r : C ! f0; 1g, is called the syndrome of the diagnosis. A subset F(V is said to be compatible with a syndrome r if r can arise from the circumstance that all vertices in F are faulty and all vertices in V n F are fault-free. A faulty comparator can lead to unreliable results, so a set of faulty vertices may produce different syndromes. Let rF ¼ frjr is compatible with F}. Two distinct subsets F 1 and F 2 of VðGÞ are said to be indistinguishable if and only if rF1 \ rF 1 – /; otherwise, both of F 1 and F 2 are said to be distinguishable. There are several different ways to verify whether a system is t-diagnosable under the comparison approach. The following lemma obtained by Sengupta and Dahbura [14] gives necessary and sufficient conditions to ensure distinguishability. Lemma 1. [14] Let G be a graph. For any two distinct subsets F 1 ; F 2 of VðGÞ. ðF 1 ; F 2 Þ is a distinguishable pair if and only if at least one of the following conditions is satisfied. (1) There are two distinct vertices u; w 2 VðGÞ ðF 1 [ F 2 Þ and there is a vertex v 2 F 1 DF 2 such that ðu; v Þw 2 C; (2) There are two distinct vertices u and v 2 F 1 n F 2 and there is a vertex w 2 VðGÞ ðF 1 [ F 2 Þ such that ðu; v Þw 2 C; or (3) There are two distinct vertices u; v 2 F 2 n F 1 and there is a vertex w 2 VðGÞ ðF 1 [ F 2 Þ such that ðu; v Þw 2 C. Lin et al. [15] introduced the so-called conditional diagnosability of a system under the situation that no set of faulty vertices can contain all neighbors of any vertex in the system. A faulty set F VðGÞ is called a conditional faulty set if N G ðv Þ Ü F for every vertex v 2 VðGÞ. A system GðV; EÞ is said to be conditionally t-diagnosable if F 1 and F 2 are distinguishable for each pair of distinct conditional faulty set F 1 and F 2 with jF 1 j 6 t; jF 2 j 6 t. The maximum value of t such that G is conditionally tdiagnosable is called the conditional diagnosability of G, denoted by t C ðGÞ. It is trivial that tC ðGÞ P tðGÞ. By the operation of 1-edges in Sn;k , there exist three vertices u; v ; w 2 VðSn;k Þ, such that ðu; v ; wÞ in a cycle of length 3, We set A ¼ N½u; v ; w; F 1 ¼ A fw; v g, and F 2 ¼ A fu; wg. We get jF 1 j ¼ jF 2 j ¼ 3ðk 1Þ þ n k 2 þ 1 ¼ n þ 2k 4, and jF 1 F 2 j ¼ jF 2 F 1 j ¼ 1. It is easy to check that F 1 and F 2 are two conditional faulty sets, and F 1 and F 2 are indistinguishable. Hence we have the following result: Theorem 5. tc ðSn;k Þ 6 n þ 2k 5 for n P 5; k P 3; n k P 2. The following two lemmas are necessary in our context. Lemma 2. Let F 1 and F 2 be any two distinct conditional fault-sets of Sn;k ðn P 5; k P 3; n k P 2Þ with jF 1 j 6 n þ 2k 5; jF 2 j 6 n þ 2k 5. Denote by H the maximum component of Sn;k F 1 \ F 2 . Then, for every vertex u 2 F 1 DF 2 ; u 2 H.
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Proof. Without loss of generality, we assume that u 2 F 1 F 2 . Since F 2 is a conditional faulty set, there is a vertex v 2 ðSn;k F 2 Þ fug such that ðu; v Þ 2 EðSn;k Þ. Suppose that u is not a vertex of H. Then v is not in H, so u and v are in one small component of Sn;k F 1 \ F 2 . Since F 1 and F 2 are distinct, we have jF 1 \ F 2 j 6 n þ 2k 6. Hence fu; v g forms a component K 2 in Sn;k F 1 \ F 2 by Theorem 3, i.e., the vertex u is the unique neighbor of v in Sn;k F 1 \ F 2 . This is a contradiction since F 1 is a conditional faulty set, but all the neighbors of v are faulty in Sn;k F 1 . h Lemma 3 [15]. Let G be a graph with dðGÞ P 2, and let F 1 and F 2 be any two distinct conditional faulty sets of G with F 2 F 1 . Then, ðF 1 ; F 2 Þ is a distinguishable conditional pair under the comparison diagnosis model. Lemma 4. Let F 1 and F 2 be any two distinct conditional faulty sets of Sn;k ðn P 5; k P 3; n k P 2Þ with jF 1 j 6 n þ 2k 5 and jF 2 j 6 n þ 2k 5. Then, ðF 1 ; F 2 Þ is a distinguishable conditional pair under the comparison diagnosis model. Proof. By Lemma 3, ðF 1 ; F 2 Þ is a distinguishable conditional pair of Sn;k if F 1 F 2 or F 2 F 1 . Thus, we assume that jF 1 F 2 j P 1, and jF 2 F 1 j P 1. Let S ¼ F 1 \ F 2 . Then we have jSj 6 n þ 2k 6. Let H be the largest connected component of Sn;k F 1 [ F 2 . By Lemma 2, every vertex in F 1 DF 2 is in H. We claim that H has a vertex u outside F 1 [ F 2 that has no neighbor in S. Since every vertex has degree n 1, the vertices in S have at most ðn 1ÞjSj neighbors in H. There are at most 2ðn þ 2k 6Þ jSj vertices in F 1 [ F 2 and at most two vertices of Sn;k S may not belong to H by Theorem 4. Since the fact is jSj 6 n þ 2k 6, we have
n! n! ð2ðn þ 2k 6Þ jSjÞ 2 P ðn 2ÞjSj 2ðn þ 2k 6Þ 2 ðn kÞ! ðn kÞ! n! P ðn þ 2k 6Þðn 2Þ 2ðn þ 2k 6Þ 2 P 4 for n P 5; k ðn kÞ! P 3; n k P 2: Thus, there must be some vertex of H outside F 1 [ F 2 , which has no neighbors in S. Let u be such a vertex. If u has no neighbor in F 1 [ F 2 , then we can find a path of length at least two within H to a vertex v in F 1 [ F 2 . We may assume that v is the first vertex of F 1 DF 2 on this path, and let q and w be the two vertices on this path immediately before v (we may have u ¼ q), so q and w are not in F 1 [ F 2 . The existence of the edges ðq; wÞ and ðw; v Þ shows that ðF 1 ; F 2 Þ is a distinguishable conditional pair of Sn;k by Lemma 1. Now we assume that u has a neighbor in F 1 DF 2 . Since the degree of u is at least 3, and u has no neighbor in S, there are three possibilities: (1) u has two neighbors in F 1 F 2 ; or (2) u has two neighbors in F 2 F 1 ; or (3) u has at least one neighbor outside F 2 [ F 1 . In each subcase above, Lemma 1 implies that ðF 1 ; F 2 Þ is a distinguishable conditional pair of Sn;k under the comparison diagnosis model, finishing the proof. h Theorem 6. tc ðSn;k Þ ¼ n þ 2k 5 for n P 5, k P 3; n k P 2. The class of alternating group networks was introduced in the late 1990’s as an alternative to the alternating group graphs as interconnection networks. Cheng et al. [30] showed that this topology is in fact isomorphic to the ðn; n 2Þ-star graph, a member of the well-known (n, k)-star graphs. Zhou and Xiao [31] derived that the conditional diagnosability of alternating group networks AN n under the comparison model. In fact, their result is a special case of theorem 6. Theorem 7 [31]. t c ðANn Þ ¼ 3n 9 for n P 5. 5. Conclusion The paper is interested in the study of fault resiliency of interconnection networks. The fault resiliency can be used to reveal the conditional connectivity of high order, and also to evaluate the conditional diagnosability under the comparison model. This method can be applied to other complex network structures. Acknowledgments The author thanks the anonymous referees for their valuable suggestions. This research was financially supported by National Natural Science Funds of China (No. 61072080) and the Key Project of Fujian Provincial Universities services to the western coast of the straits-Information Technology Research Based on Mathematics.
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