#### Abstract

In the research on the reliability of a connection network, diagnosability is an important problem that should be considered. In this article, a new concept regarding diagnosability, called strong local diagnosability (SLD), which describes the local status of the strong diagnosability (SD) of a system, is presented. In addition, a few important results related to the SLD of a node of a system are presented. Based on these results, we conclude that in a hypercube network of dimensions, denoted by , the SLD of a node is equal to its degree when . Moreover, we explore the SLD of a node of an incomplete hypercube network. We determine that the SLD of a node is equal to its remaining degree (RD) in an incomplete hypercube network, which is true provided that the number of faulty edges in this hypercube network does not exceed . Finally, we discuss the SLD of a node for an incomplete hypercube network and obtain the following results: if the minimum RD of nodes in an incomplete hypercube network of -dimensions is greater than , then the SLD of any node is still equal to its RD provided that the number of faulty edges does not exceed . If the RD of each node is greater than , then the SLD of each node is also equal to its RD, no matter how many faulty edges exist in .

#### 1. Introduction

In a multiprocessor system incorporating a large number of processors (nodes), some processors may fail. In other words, there may exist faulty processors in such a system. Faulty processors in a multiprocessor system will affect the reliability of the system. Hence, in the system design, the problem of self-diagnosis should be considered. Whenever a faulty node is identified by the system, it should be repaired or replaced with an additional one. The process of identifying faulty processors is referred to as fault diagnosis, which has been widely studied in [1–9]. The diagnosability, a key measurement of self-diagnosis capacity, of a system is the upper limit on the number of faulty processors that a system is certain to identify.

The hypercube network structure [10] is a popular topology in modeling a multiprocessor system. An -dimensional hypercube network, denoted as , consists of nodes and edges. A binary -bit string, e.g., , can be employed to denote the address of a node in . If the address of node and the address of node are different in exactly one bit position, then they are connected by a link, and vice versa. For two nodes and in , if and are connected by a link and their addresses differ in the th position, then this edge is called the th-dimensional edge. can be decomposed in two -dimensional hypercube networks and along the th-dimensional edges. Let . Then there must be another node satisfying . We call this edge the crossing edge between and .

In system-level fault diagnosis, two models that are typically used as fault diagnosis models are the PMC model [11] and the comparison model [12]. The fault diagnosis model in this paper is the PMC model. “PMC” stands for Preparata, Metze, and Chien, as this model was first proposed by Preparata et al. in [11]. In [11], Preparata et al. put forward the concept of a one-step -diagnosable system and of a sequentially -diagnosable system. Since then, results corresponding to the PMC model have been widely reported (see [9, 11, 13–19]). In the PMC model, when we consider the diagnosability of a system with missing edges, if all neighbors of some node are faulty, then it is impossible to judge whether or not the node is fault-free. In other words, in the PMC model, the diagnosability of a system with missing edges depends on the upper limit of the minimum degree of . In practice, the probability that all neighbors of a node are synchronously faulty is very low. For example, consider an -dimensional hypercube network ; the number of node subsets such that its cardinality is is . However, the number of node subsets such that its size is and it incorporates all neighbors of some node is at most . The ratio implies that the probability that there exists some node such that its neighbors are all faulty is very low. For this reason, Lai et al. [14] proposed a concept called a strongly -diagnosable system (STDS). Lai et al. [14] claimed that a STDS is a -diagnosable system provided that the system does not have a node such that its neighbors are all faulty simultaneously, the probability of which is very low. In other words, a STDS is nearly a -diagnosable system. In a certain sense, the SD of a system is more precise than its traditional diagnosability. Much research on SD has been carried out (see [2–4, 6, 9, 14]).

It is worth mentioning that in analyzing the diagnosability of a system, if one considers only the global situation and ignores some local information, it is possible that the diagnosability obtained is not the largest. For example, let denote the integrating system produced by and , where . The diagnosability of may be less than . However, the system may diagnose all faulty processors even if their number is greater than and less than . Thus, it is possible that if some local details of a system are ignored, then the amount of details obtained will be less than that in the practical case. In practice, the local status of a system is also an indicative of the entire system. Hence, when we consider the problem of the diagnosability of a system, it is necessary to study the local properties of the system. In 2007, Hsu and Tan [20] proposed a novel diagnosability, called local diagnosability, in order to study the problem of fault diagnosis of a system under the PMC model. Later, in 2009, Chiang and Tan [13] extended the results in [20] from the PMC model to the comparison model. Now, we are led to the following question: is it possible to present a concept that describes the local status with respect to the SD of a system? This article presents a concept called SLD, which describes nodes’ local SD status of a system. A sufficient and necessary condition, which tests the SLD of a node and its relationship with the SD of a system, is discussed in this article. Based on this sufficient and necessary condition, we conclude that the SD of a system can be determined by computing the SLD of each node of the system. Then, it is obtained that the SLD of a node of is , where . Moreover, we discuss the SLD of a given node of an incomplete hypercube network [21].

The remainder of this paper is as follows: after introducing some necessary preliminaries and the terminology used in this paper, in Section 3, we present the concept of SLD and an important theorem for checking the SLD of a given node in a system. In Section 4, the SLD of a node of is studied. Finally, in Section 5, we discuss the conclusions of our paper.

#### 2. Preliminaries

A system is usually described by a graph , where represents all communication edges between processors and represents the set of all processors. Throughout this paper, we will use the following terms interchangeably: graph, multiprocessor system and interconnection system. Moreover, we will use node, vertex, and processor interchangeably. The definition of a graph follows that given in [22]. For a node in , a node is said to be its neighbor if . Moreover, let (, in no confusion) represent the set of all neighbors of . The degree of a node in , denoted by (, no confusion), is the cardinality of , i.e., . If the degree of each node in is , then is said to be -regular. For a disconnected graph , a connected component of refers to a maximal connected subgraph of . If a connected component does not have any edges, it is said to be trivial; otherwise, it is nontrivial. Given a node subset , let represent a subgraph created by deleting from . Similarly, given an edge subset , let represent a graph obtained by removing from . The connectivity of , denoted by , is the cardinality of the minimum node set () such that is disconnected. If , then is said to be -connected. For a graph , we use (, respectively) to denote the node set of (the edge set of , respectively).

In the PMC model, if , then and can be used to test each other. We use the order pair to represent that tests . In this situation, is the tester, and is the tested node. The PMC model assumes that if is diagnosed to be faulty (fault-free) by , then the outcome of testing is 1 (0), denoted by (). We use to denote a syndrome. The collection of all faulty nodes in is called a faulty set. It is possible that any subset of is a faulty set. The fault diagnosis of a system refers to the process of identifying all faulty nodes in the system. In a system , the cardinality of a maximum faulty node set that can be identified by is called its diagnosability, denoted as . Given a syndrome , a set is called consistent with if the following condition is true: if and , then if and only if .

We use to denote the set of syndromes produced by faulty node set . Two subsets are called distinguishable if . If and are distinguishable, then is said to be a distinguishable pair; otherwise, is said to be an indistinguishable pair. Let . In addition, we need some previous results concerning the -diagnosable system. Throughout this paper, the fault diagnosis model refers to the PMC model.

Lemma 1 (see [23]). *A system is -diagnosable if and only if for two arbitrary different subsets satisfying , is always a distinguishable pair.*

Lemma 2 (see [23]). *In a system , suppose that with are two different subsets. Then is a distinguishable pair if and only if has at least an edge such that and (Figure 1).*

**(a)**

**(b)**

The following lemmas are related to the concept of node diagnosability:

Lemma 3 (see [20]). *Suppose that is a system, . is locally -diagnosable at if and only if for each subset with , , and , the number of nodes in the connected component containing in is greater than .*

Lemma 4 (see [20]). *Suppose that is a graph, with . Then, the locally diagnosability (LD) of node does not exceed .*

Lemma 5 (see [20]). *In a system , suppose that . Then the system is -diagnosable at node if and only if for two arbitrary different subsets satisfying and , has at least an edge such that and .*

Lemma 6 (see [20]). *A -diagnosable system is locally -diagnosable at each node. Conversely, if is locally -diagnosable at each node, then is -diagnosable.*

By Lemma 1 and Lemma 5, we conclude that a -diagnosable system must be -diagnosable at each node, and vice versa.

*Definition 1. *A system is triangle-free if it does not incorporate a triangle.

*Remark 1. *Among the regular interconnection networks, there are many famous networks that are triangle-free, for example, hypercube-like networks [18], star networks [7], and the exchanged hypercube network [24].

The following definition of a strongly -diagnosable system follows [14].

*Definition 2. *A -diagnosable system, denoted by , is called a strong -diagnosable system if the following condition is true.

For any two different subsets with and , if is an indistinguishable pair, then has at least a node such that .

The strong diagnosability (SD) of a system is the maximum number value of such that the system is a strong -diagnosable system.

#### 3. The Strong Local Diagnosability

First, let us look back at some conclusions regarding diagnosability. Reference [20] proposed a new concept, called node diagnosability, that describes the local status with respect to system diagnosability. By Lemma 6, to obtain the system diagnosability, we need only to compute the node diagnosability of each node of the system, which is a novel way to study the system diagnosability. By Definition 2, we have that a STDS must be a -diagnosable system but may not be a -diagnosable system [13]. Since the probability that all neighbors of some node are faulty simultaneously is rather small, a STDS is almost -diagnosable. Motivated by this strategy, we propose a new concept called the SLD at a given node in a system. This new concept combines the characteristics of node diagnosability and SD. For a system , , we use (; ) to denote the diagnosability of (the LD at node in ; the SLD at node in ).

*Definition 3. *Let be the diagnostic graph of a system and . is called strong locally -diagnosable at node if it is locally -diagnosable at node and the following condition is true.

For any two different subsets with , and , if is an indistinguishable pair, then .

A system is said to be strong locally -diagnosable if for each node in , is strong locally -diagnosable at node .

Proposition 1. *Suppose that is a graph, with . The SLD of node in does not exceed .*

*Proof 1. *By Lemma 4 and Definition 3, the proof can be easily obtained.

In the following, we propose two propositions for describing the relationship between a strong locally -diagnosable system and a strongly -diagnosable system:

Proposition 2. *If is strong locally -diagnosable, then is strongly -diagnosable.*

*Proof 2. *Suppose that is not strongly -diagnosable. From Definition 1, it is obtained that there is an indistinguishable pair with , , and such that for any node , . Since , there exists a node such that and . Hence, is not strong locally -diagnosable at node , a contradiction to the postulation that is strong locally -diagnosable. This completes the proof.

The above proposition proposes a sufficient but unnecessary condition to test if the system is strongly -diagnosable. Next, a necessary and sufficient condition for testing the strong -diagnosability of a triangle-free system is presented.

Proposition 3. *Let be a triangle-free system and be an integer. If is strongly -diagnosable, then is strong locally -diagnosable, and vice versa.*

*Proof 3. *By the proof of Proposition 2, the sufficiency holds.

*Necessity 1. *Let be a strongly -diagnosable system; then, is -diagnosable. Let . By Lemma 6, we have that is locally -diagnosable at vertex . Next, by Definition 3, we need to prove only that for any two different subsets with and , if is an indistinguishable pair, then has at least a node and . In contrast, assuming that , we derive a contradiction. By Definition 2, there is at least one node such that . By Lemma 4, we have that . Note that , , and ; therefore, . Thus, . On the other hand, implies that or . Here, we need to consider only the situation where , as the proof of the situation in which can be similarly obtained. In the rest of this proof, four situations are considered according to the subjections of node : (1) , (2) , (3) , and (4) .

*Case 1 . *Since and , , which is a contradiction.

*Case 2 . *It is clear that . is an indistinguishable pair, implying that . Moreover, by , we have that . This is a contradiction to .

*Case 3 . *Since and , , which is a contradiction.

*Case 4 . *By the hypothesis, we have that and . We first discuss the situation where . Suppose that . By the proof of Case 2, we have that is adjacent to and that there is no edge between and any node in . (as is an indistinguishable pair) and (Lemma 4), we have that . Since is a triangle-free graph, . Thus, the degree of is no greater than 2, a contradiction to (Lemma 4). Second, we consider the situation where . Since there is no edge between and any node in , ; thus, , and therefore, . On the other hand, since , , and hence, , which is a contradiction. This completes the proof.

Proposition 2 describes the relationship between SLD and SD for a triangle-free graph. The following theorem follows Proposition 2.

Theorem 1. *In a triangle-free network system , the strong diagnosability of is equal to .*

The following sufficient and necessary condition can be applied.

Theorem 2. *In a graph , . Thus, is strong locally -diagnosable at node if and only if, for any , , , and , the following conditions hold:
*Condition 1*If , then the number of nodes of the connected component containing in exceeds .*Condition 2*If , then either (a) the number of nodes of the connected component containing in exceeds 2 or (b) .*

*Proof 4 (sufficiency). *Let denote the connected component containing in . By Condition 1, for , we have that . According to Lemma 3, is locally -diagnosable at node . Next, we need to prove that for any two different subsets with , and , if is an indistinguishable pair, then . Suppose that there exist two different subsets with , and such that is an indistinguishable pair and . Let ; then, . If , by Condition 1, the number of nodes of the connected component containing in is greater than . Moreover, the number of nodes of the connected component containing in is greater than , which implies that has two nodes and such that . Hence, is a distinguishable pair, which is a contradiction. Thus, , which implies that . By Condition 2, we have that the number of nodes of the connected component containing in exceeds 2. Hence, has two nodes and such that . Hence, is a distinguishable pair, which is a contradiction.

Next, we consider the situation . From last paragraph, we have that is locally -diagnosable at node . By Definition 3, we only need to prove that for any two different subsets with , and , if is an indistinguishable pair, then . To the contrary, assume that there exist two different subsets with , , and is an indistinguishable pair, but . Let and , then , and then we have that . Moreover, by Condition 1, we have that , which implies that . Since is an indistinguishable pair, there is no edge from to . So, the number of nodes of the connected component containing is no more than . Moreover, , a contradiction.

*Necessity 2. *We consider Condition 1 first. Suppose that for some , has a connected component containing , e.g., , such that . Let be decomposed in two subsets as follows: and , with and . Let and . It is clear that , and . Since is a connected component of , does not have an edge such that and . By Lemma 2, is an indistinguishable pair. Since is strong locally -diagnosable at node , . Hence, . On the other hand, the assumption that is strong locally -diagnosable at node implies that is -diagnosable at node . By Lemma 4, we have that , which is a contradiction. Hence, Condition 1 is necessary.

Now, let us consider Condition 2. Suppose that . Let denote the connected component containing in . If the number of nodes in is less than or equal to 2, then can be decomposed in two subsets and satisfying the following conditions: and , with and . Let and . Then, , and . By the assumption that is strong locally -diagnosable at node , we have that is locally -diagnosable at node . Moreover, we have that is an indistinguishable pair. By Definition 3, we conclude that . This completes the proof of necessity.

Corollary 1. *Suppose that is a subgraph of , . If is strong locally -diagnosable at and , then is strong locally -diagnosable at .*

*Proof 5. *For any , , , and . We will prove that the two conditions in Theorem 2 hold. Let , . Consider the following cases:

* Case 5 . *Note that . Since is strong locally -diagnosable at , the number of nodes of the connected component containing in exceeds , which implies the number of nodes of the connected component containing in exceeds . Moreover, the number of nodes of the connected component containing in exceeds . So, Condition 1 of Theorem 2 holds.

*Case 6 .*

* Case 7 . *An argument being similar to Case 5 can be used to obtain the following result: the number of nodes of the connected component containing in exceeds , which implies that the number of nodes of the connected component containing in exceeds 2. So, Condition 2 of Theorem 2 holds.

* Case 8 . *Then . Since is strong locally -diagnosable at and , then either (a) the number of nodes of the connected component containing in exceeds 2, which implies the number of nodes of the connected component containing in exceeds 2, or (b) , which implies . So, Condition 2 of Theorem 2 holds.

Next, we present the type I structure, which is used to verify the SLD of a given node.

*Definition 4. *Suppose that denotes a diagnostic graph of a system and that is a node in . The type I structure of the node can be decomposed in three node sets (, , and ) and two edge sets (, ), as shown in Figure 2. These sets are defined as follows: , , and . , and . Let () represent the number of nodes (edges) in the type I structure.

#### 4. The Hypercube Network and Incomplete Hypercube

Regular topology structures are usually used to imitate multiprocessor systems. There is no doubt that the hypercube structure is one of the most important regular topology structures. In the following, we will discuss the problem of the SLD of a hypercube network.

Theorem 3. * is strong locally -diagnosable at each node, where .*

* Proof 6. *Let be an arbitrary node. For each , , , and , we show that the two conditions of Theorem 2 are both true.

* Case 9 . *Noting that the connectivity of is , we have that is connected. Let denote this unique connected component in ; then, . Hence, (), which implies that Condition 1 of Theorem 2 holds.

* Case 10 . *Consider the type I structure of node in . If , then , which implies that Condition 2 of Theorem 2 is true. If , then there are two nodes satisfying , and . Hence, has a connected component, which incorporates and has 3 or more nodes. This completes the proof of Theorem 3.

According to Theorem 3, the SLD of every node of is the same as its degree, with . It is natural to consider the following question: is the result still true for an incomplete hypercube network? In the following, we discuss this problem. We use to denote an edge subset and to denote an incomplete hypercube network of dimensions created by removing from the hypercube network . In the following, we show that the SLD of every node is the same as its RD in , even if the cardinality of can reach . We now give an example to explain that the result is not true if the cardinality of is . In Figure 3, . Let , where and are two neighbors of ; then, . Let and ; then, , . By Lemma 2, is an indistinguishable pair, and . Hence, the SLD of node differs from its RD in .

Theorem 4. *Let be an edge subset with , where . Let denote the induced subgraph of . Then, the SLD of every node of is exactly the same as its RD.*

*Proof 7. *We prove this theorem by induction . First, let us consider the situation where ; in this case, . For , based on Theorem 3, the result holds. Assume that . We consider the type I structure of node in , as shown in Figure 4. It is clear that . If , then the RD of is 3. We can use a similar argument to that used in the proof of Theorem 3 to show that the SLD of node is 3. If , then the RD of is 4. Let and , with and . When and , Condition 2 of Theorem 2 holds. When and , consider the following cases.

* Case 11 . *Then the number of nodes of the connected component containing in is at least 3, Condition 2 of Theorem 2 holds.

* Case 12 . *Then . Let , then has at least 2 nodes, which implies . So, the number of nodes of connected component containing in is at least 3. In other words, Condition 2 of Theorem 2 holds.

Next, we show that when , , where is the connected component containing in . From Figure 5, we conclude that the only case stopping the condition from being satisfied is that in which and . However, based on the definition of a hypercube network, the connected component contains other neighbors of , e.g., and , as well as their neighbors, which do not include . Hence, for connected component , it is true that when . In summary, for , the result is true.

Assume that when for , the result is true. Next, we prove that for , and , the result is still true. If , then the result holds. Now, we consider the situation where . Decompose in two- dimensional hypercube networks and along the th-dimensional edges. Let , . Let be an arbitrary node. Without the loss of generality, let and . Suppose that is a crossing edge, namely, is one of the th-dimensional edges, where . Consider the following two cases:

*Case 13 . *Then, . By the inductive hypothesis, we have that . On the other hand, we have that . According to Corollary 1 and Proposition 1, we have that . Hence, , and thus the result is true.

*Case 14 . *Then, . By Proposition 1, we have that . Now, we need to prove only that is strong locally -diagnosable at node . Let , with , , , and . Next, we prove that the two conditions of Theorem 2 hold. For the sake of convenience, let stand for the connected component containing in , denote the connected component containing in , and denote the connected component containing in ; moreover, let .

* Case 15. *First, we consider the condition that . We will prove that . Consider the following cases:

*Case 16 .*

* Case 17 . *If , then . Moreover, . If , then . Moreover, .

* Case 18 . *It is obvious that . Let ; then, has at least one node, e.g., , which is not . Let ; then, .

If , let denote the connected component containing in . Since is the subgraph of , . Note that ; by inductive hypothesis and Theorem 2, we have that . In , we can choose two distinct nodes and such that either both and are neighbors of or is a neighbor of and is a neighbor of . Suppose that and are two crossing edges in , where and . If , then , and then . If , then , and then .

If , then . Since , there must exist one neighbor of , e.g., , such that in . Let be a crossing edge in , where . Let denote the connected component containing in ; then, . Note that ; we have that belong to a connected component of , which implies that .

*Case 19 .*

If , then . Since the connectivity of is , . Hence, . If , then either or . Hence, .

* Case 20 . *Next, we show that . Since , the degree of any node in is at least 3. Therefore, has two neighbors, e.g., and , in . Consider the following cases:

* Case 21 . *By inductive hypothesis and Theorem 2, we have that . If , then . If , then .

* Case 22 . *Since , there exists another neighbor of , e.g., , in . If , then , and then . If , then . Hence, by inductive hypothesis and Theorem 2, we have that .

* Case 23. *Next, we consider the condition that . Suppose that . Consider the following cases.

* Case 24 . *Since , there is at least node in such that . Hence, belong to a connected component in . Condition 2 of Theorem 2 is satisfied.

* Case 25 . *By , we conclude that there exists at least node in , e.g., , such that . Since , then there is at least one node in , e.g., , such that . Then, belong to a connected component in . Condition 2 of Theorem 2 is satisfied.

In summary, is strong locally -diagnosable at the node .

Clearly, the following corollary follows Theorem 4; hence, its proof is omitted.

Corollary 2. *For and with , is strongly -diagnosable, where represents the minimum degree of nodes in .*

In the previous section, we concluded that the SLD of a node is its RD in an incomplete hypercube network created by removing the edge set with from . In the following, we discuss the problem of the upper bound of such that the following two conditions are satisfied: (1) the RD of each node in , denoted by , is at least 4; (2) the SLD of every node is the same as its RD. Now, we consider an example in which the SLD of a node in differs from its RD in with , even though the minimum degree of nodes in is more than 3. Assume that , the adjacent situations of which are shown in Figure 6. Let , , and ; then, . Let and . Since , is an indistinguishable pair; at the same time, , is not , and the RD of is in .

Theorem 5. *Let , where with . Assume that , then , it follows that .*

*Proof 8. *Before proving this theorem, we define some notions for further discussion. Consider another structure of a node obtained by extending its type I structure, called the type I-EX structure of node , in the following way: insert a node set and an edge set in the type I structure of node , where and . After removing from , the type I-EX structure of node becomes another new structure, called the type II structure of node . The set of nodes (; ; ) becomes another new set of nodes (; ; ) in the type II structure of node