Intermittent Fault Diagnosability of Hyper Petersen Network
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Abstract
The problem of permanent fault diagnosis has been discussed widely, and the diagnosability of many wellknown networks have been explored. Faults of a multiprocessor system generally include permanent and intermittent, with intermittent faults regarded as the most challenging to diagnose. In this paper, we investigate the intermittent fault diagnosability of hyper Petersen networks. First, we derive that an \(n\)dimensional hyper Petersen network \(HP_{n}\) with faultfree edges is \((n  1)_{i}\)diagnosable under the PMC model. Then, we investigate the intermittent fault diagnosability of \(HP_{n}\) with faulty edges under the PMC model. Finally, we prove that an \(n\)dimensional hyper Petersen network \(HP_{n}\) is \((n  2)_{i}\)diagnosable under the MM* model.
Keywords
Fault diagnosability Intermittent fault PMC model Hyper Petersen network Multiprocessor system1 Introduction
In recent years, large multiprocessor systems have become increasingly popular due to continuing advancements in semiconductor technology. A multiprocessor system contains hundreds of thousands of processors (nodes). However, some nodes of a multiprocessor system may be faulty while the system is in operation. To maintain the reliability of a multiprocessor system, faulty nodes need to be identified and replaced by faultfree nodes. Fault diagnosis is the process of identifying faulty nodes in a multiprocessor system. The diagnosability of a system is the maximum number of faulty nodes that are guaranteed to be identified. Therefore, diagnosability is an important metric for measuring the reliability of multiprocessor systems.
Several approaches have been developed for diagnosing faulty nodes in a multiprocessor system. One important approach, the PMC model, was proposed by Preparata et al. [1]. However, the diagnostic capabilities of classical diagnostic strategies under the PMC model are limited [2, 3, 4]. To enhance the system’s selfdiagnosing capability, some new diagnosis strategies have been proposed [5, 6, 13]. The conditional diagnosability [5] is introduced through the restriction that no faulty set can contain all the neighbors of any node in a graph. Peng et al. [6] defined a related metric, the ggoodneighbor conditional diagnosability. The t/kdiagnosis [13] was proposed by Somani and Peleg, which guarantees that all the faulty nodes (processors) in a system are detected (provided the number of faulty units does not exceed \(t\)) while, at most, \(k\) nodes are incorrectly diagnosed. Examples of important advances by these diagnostic strategies may be found in [7, 8, 9, 10, 11, 12]. The comparison model is another popular system diagnosis model [14, 15]. Each test involves three nodes under the comparison model. One node acts as a comparator. It sends test tasks to the other two nodes and outputs the corresponding diagnosis result by comparing whether the two returned results are consistent. Some representative progresses under the comparison model may be found in [16, 17, 18, 19].
In these previous works, there is an important assumption that all faults are permanent. However, faults of a multiprocessor system include not only permanent, but also intermittent. An intermittent failure is the loss of some function or performance characteristic in a product for a limited period of time and subsequent recovery of the function [20]. Historically, intermittent faults were considered as a prelude to permanent faults [21]. Consequently, they have received much attention [22, 23]. Intermittent faults have long been recognized as a highly important source of failures within multiprocessor systems. However, diagnosis of intermittent faults is more challenging because intermittent faults are more difficult to detect than permanent faults. To illustrate the difference between intermittent fault diagnosis and permanent fault diagnosis, we show a simple example in Sect. 2.
The ndimensional hypercube \(Q_{n}\) is one of the most popular interconnection networks for multiprocessor interconnection. The hyper Petersen network \(HP_{n}\) [25, 26] is proposed as an attractive variation of the ndimensional hypercube \(Q_{n}\) which has desirable properties such as high symmetry, high connectivity, and logarithmic diameter. The hyper Petersen network \(HP_{n}\) is an a \(n\)regular graph.
In this paper, we investigate the intermittent fault diagnosability of the ndimensional hyper Petersen network under the PMC model and under the MM* model. We prove that the intermittent fault diagnosability of an ndimensional hyper Petersen network \(HP_{n}\) is \(n  1\) under the PMC model. Moreover, we discuss the intermittent fault diagnosability of an ndimensional hyper Petersen network \(HP_{n}\) in the presence of arbitrary distributed faulty edges under the PMC model. In addition, We prove that the intermittent fault diagnosability of an ndimensional hyper Petersen network \(HP_{n}\) is \(n  2\) under the MM* model.
The remainder of this paper is organized as follows. In Sect. 2, we present definitions, notation, and terminology. Then, in Sect. 3, we investigate the intermittent fault diagnosability of an ndimensional hyper Petersen network under the PMC model. Intermittent fault diagnosability of an \(n\)dimensional hyper Petersen network \(HP_{n}\) under the MM* model is also introduced in Sect. 4. Finally, conclusions are presented in Sect. 5.
2 Preliminaries
The multiprocessor system consisting of \(n\) processors is often modeled by a graph \(G = \left( {V,E} \right)\) with the nodes representing processors, and the edges representing links between the nodes. The sets of nodes and edges in \(G\left( {V,E} \right)\) are written as \(V\left( G \right)\) and \(E\left( G \right)\), respectively. The cardinalities of \(V\left( G \right)\) and \(E\left( G \right)\) are denoted by \(\left {V\left( G \right)} \right\) and \(\left {E\left( G \right)} \right\), respectively. The neighborhood set of a node \(v \in V\left( G \right)\) is the set of all nodes that are adjacent to \(v\) in \(G\left( {V,E} \right)\), denoted by \(N_{G} \left( v \right)\), and abbreviated as \(N\left( v \right)\) without ambiguity. The cardinality \(\left {N_{G} \left( v \right)} \right\) is called the degree of \(v\) in \(G(V,E)\), denoted by \({ \deg }_{G} ( v)\), and abbreviated as \({ \deg }( v)\) without ambiguity. Similarly, the set \(N\left( U \right)\) can be defined for a neighborhood set of a node set \(U \subseteq V\left( G \right)\) as follows: \(N(U) = \bigcup\nolimits_{v \in U} {N(v)  U}\). Given an edge set \(U\) of \(E\left( G \right)\), the notation \(G  U\) is used to represent the subgraph of \(G\) obtained by deleting all the edges in \(U\) from \(G\).
Suppose that all the faults are permanent, a faultfree node always correctly evaluates the tested node as being faulty or faultfree. In the PMC model [1], a node sends a task to its neighbors and then evaluates the neighbors’ responses, where all adjacent nodes are able to perform tests on each other. For two adjacent nodes \(u,v \in V\left( G \right)\), the ordered pair \(\left( {u,v} \right)\) represents the test performed by \(u\) on \(v\), where \(u\) is called the tester and \(v\) is called the tested node. The test outcome by a faultfree tester is 1 (respectively, 0) if the tested node is faulty (respectively, faultfree), denote by \(a_{uv} = 1\) (respectively, \(a_{uv} = 0\)). If the tester is faulty, then the outcome is unreliable. The collection of all test outcomes over the entire system is referred to as a syndrome \(\sigma\). A syndrome \(\sigma\) is considered to be compatible with a faulty set \(F\) if and only if \(u \in V  F\), \(\sigma \left( {u,v} \right) = 1\) implies \(v \in F\).
 1.
\(\sigma (u_{2} ,u_{3} )_{{u_{1} }} = 0\), if \(u_{1} ,u_{2} ,u_{3} \in R\);
 2.
\(\sigma (u_{2} ,u_{3} )_{{u_{1} }} = 0\) or 1, if \(\{ u_{1} ,u_{2} ,u_{3} \} \bigcap {F = \phi }\);
In contrast to permanently faulty nodes, nodes with intermittent failures are not always in a fault state. The evaluation of the tested faulty node may be incorrect if a faultfree node has not tested the faulty node sufficiently; i.e., the faultfree testing node might evaluate the faulty tested node as being faultfree. Accordingly, it might be necessary to test that node several times to achieve a correct evaluation. More details on the process of intermittent fault diagnosis can be found in [22].
The following lemma describes a necessary and sufficient condition for a multiprocessor system \(G = \left( {V,E} \right)\) with intermittent failures to be \(t_{i}\)diagnosable under the PMC model.
Lemma 2.1
[22] A system \(S\) is \(t_{i}\)fault diagnosable under the PMC model if and only if, given any 2 sets of nodes in the system, \(S_{1}\) and \(S_{2}\), \(\left {S_{1} } \right\), \(\left {S_{2} } \right \le t_{i}\), \(S_{1} \bigcap {S_{2} = \varPhi }\), the set \(R = V\left( S \right)  S_{1} \bigcup {S_{2} }\) of the remaining nodes is such that both \(S_{1}\) and \(S_{2}\) receive at least one testing link from \(R\).
Definition 2.1
[25]. The 3dimensional hyper Petersen network \(HP_{3}\) is the regular Petersen graph. Let \(PM\) be an arbitrary perfect matching between \(HP_{n  1}^{0}\) and \(HP_{n  1}^{1}\), where \(V\left( {HP_{n  1}^{0} } \right) = \{ 0v_{n + 3} v_{n + 2} \cdot \cdot \cdot v_{0} v_{n + 3} v_{n + 2} \cdot \cdot \cdot v_{0} \in V\left( {HP_{n  1} } \right)\}\) and \(V\left( {HP_{n  1}^{1} } \right) = \{ 1v_{n + 3} v_{n + 2} \cdot \cdot \cdot v_{0} v_{n + 3} v_{n + 2} \cdot \cdot \cdot v_{0} \in V\left( {HP_{n  1} } \right)\}\). For \(n \ge 4\), the \(n\)dimensional hyper Petersen network, denoted by \(HP_{n}\), can be constructed by connecting \(HP_{n  1}^{0}\) and \(HP_{n  1}^{1}\) via \(PM\).
Figure 2 illustrates \(HP_{3}\) and \(HP_{4}\).
3 Intermittent Fault Diagnosability of \(HP_{n}\) Under the PMC Model
In this section, we investigate the diagnosability of a hyper Petersen network containing intermittent faulty nodes under the PMC model. For the following discussion, an important property of hyper Petersen networks is presented.
Lemma 3.1
Let \(HP_{n} \left( {n \ge 3} \right)\) be an \(n\)dimensional hyper Petersen network. Let \(U\) be a node set of \(HP_{n}\), \(U \subseteq V\left( {HP_{n} } \right)\). Then, \(\left {N_{{HP_{n} }} \left( U \right)} \right \ge n\) if \(1 \le \left U \right \le n  1\).
Proof

Case 1 \(U \subseteq V\left( {HP_{n}^{0} } \right)\) or \(U \subseteq V\left( {HP_{n}^{1} } \right)\)
Let \(U \subseteq V\left( {HP_{n}^{0} } \right)\), we divide the proof into two subcases according to different values of \(\left U \right\). The first subcase is \(1 \le \left U \right \le n  2\); the second subcase is \(\left U \right = n  1\).

Subcase 1.1 \(1 \le \left U \right \le n  2\).
By the induction hypothesis, \(\left {N_{{HP_{n}^{0} }} \left( U \right)} \right \ge n  1\) because \(1 \le \left U \right \le n  2\). According to the definition of hyper Petersen network \(HP_{n}\), each node of \(U\) has a neighbor node in \(HP_{n}^{1}\), i.e., \(\left {N_{{HP_{n} }} \left( U \right)\bigcap {V\left( {HP_{n}^{1} } \right)} } \right \ge 1\). Hence, \(\left {N_{{HP_{n} }} \left( U \right)} \right \ge n\).

Subcase 1.2 \(\left U \right = n  1\).
Similar to the above discussion, each node of \(U\) has a neighbor node in \(HP_{n}^{1}\); thus, \(U\) has \(n  1\) neighbor nodes in \(HP_{n}^{1}\), \(\left {N_{{HP_{n} }} \left( U \right)\bigcap {V\left( {HP_{n}^{1} } \right)} } \right = n  1\). Let \(v \in U\), there are \(n  1\) neighbor nodes of \(v\) in \(HP_{n}^{0}\). Because \(v \in U\) and \(\left U \right = n  1\), \(v\) must have at least one neighbor node \(w\) in \(HP_{n}^{0}\), \(w \notin U\), i.e., \(\left {N_{{HP_{n} }} \left( U \right)\bigcap {V\left( {HP_{n}^{0} } \right)} } \right \ge 1\). Accordingly, \(\left {N_{{HP_{n} }} \left( U \right)} \right \ge n\).
Similarly, this lemma holds for \(U \subseteq V\left( {HP_{n}^{1} } \right)\).

Case 2 \(\left {U\bigcap {V(HP_{n}^{0} )} } \right > 0\) and \(\left {U\bigcap {V\left( {HP_{n}^{1} } \right)} } \right > 0\), with \(\left {U\bigcap {V\left( {HP_{n}^{0} } \right)} } \right + \left {U\bigcap {V\left( {HP_{n}^{1} } \right)} } \right \le n  1\)
Since \(0 < \left {U\bigcap {V\left( {HP_{n}^{0} } \right)} } \right \le n  2\), \(\left {N_{{HP_{n} }} \left( U \right)\bigcap {V\left( {HP_{n}^{0} } \right)} } \right \ge n  1\) by the induction hypothesis. Similarly, \(\left {N_{{HP_{n} }} \left( U \right)\bigcap {V\left( {HP_{n}^{1} } \right)} } \right \ge n  1\). Accordingly, \(\left {N_{{HP_{n} }} \left( U \right)} \right \ge 2\left( {n  1} \right) \ge n\left( {n \ge 3} \right)\)
This completes the proof of this Lemma.
According to Lemma 3.1, for an \(n\)dimensional hyper Petersen network, the cardinality of the neighbors set of the set \(S\) is not less than \(n\) if the cardinality of the set \(S\) is less than \(n\). Since a hyper Petersen network \(HP_{n}\) is an \(n\)regular graph, we can construct the structure of Lemma 2.1 by Lemma 3.1.
Theorem 3.1
An \(n\)dimensional hyper Petersen network \(HP_{n}\) is \((n  1)_{i}\)diagnosable for \(\left( {n \ge 3} \right)\).
Proof
Let \(S_{1}\) and \(S_{2}\) be two node sets of \(HP_{n}\), with \(\left {S_{1} } \right\), \(\left {S_{2} } \right \le n  1\), and \(S_{1} \bigcap {S_{2} = \varPhi }\). For \(S_{1}\), \(\left {N_{{HP_{n} }} \left( {S_{1} } \right)} \right \ge n\) by Lemma 3.1. \(\left {N_{{HP_{n} }} \left( {S_{1} } \right)\left  \rightS_{2} } \right \ge 1\). Thus, there is one edge from \(V\left( {HP_{n} } \right)  S_{1} \bigcup {S_{2} }\) to \(S_{1}\). Similarly, there is one edge from \(V(HP_{n} )  S_{1} \bigcup {S_{2} }\) to \(S_{2}\). By Lemma 2.1, the \(n\)dimensional hyper Petersen network \(HP_{n}\) is \((n  1)_{i}\)diagnosable. The proof is complete.
In the above discussion, we showed that an \(n\)dimensional hyper Petersen network \(HP_{n}\) is \((n  1)_{i}\)diagnosable when all edges of \(HP_{n}\) are working. Since the multiprocessor system consists of hundreds of thousands of communication links, avoiding failure of some communication links is difficult in some cases. Therefore, we are led to the following question: what is the intermittent fault diagnosability of hyper Petersen networks with faulty edges? First, we give some properties of \(HP_{n}\) for the following discussion; then, we show that \(HP_{n} \left( {n \ge 3} \right)\) is \((r  1)_{i}\)diagnosable if \({ \hbox{min} }\{ { \deg }\left( v \right)v \in V\left( {HP_{n} } \right)\} = r\) and \(2 \le r \le n\).
Lemma 3.2
Let \(HP_{n} \left( {n \ge 3} \right)\) be an \(n\)dimensional hyper Petersen network. Then \(HP_{n}\) has no cycle of length 3.
Proof

Case 1 \(\left\{ {u,v,w} \right\} \subseteq V\left( {HP_{n}^{0} } \right)\) or \(\left\{ {u,v,w} \right\} \subseteq V\left( {HP_{n}^{1} } \right)\).
According to the hypothesis that both \(HP_{n}^{0}\) and \(HP_{n}^{1}\) have no cycle of length 3, this case is impossible.

Case 2 Without loss of generality, assume that \(\left\{ {u,v} \right\} \subseteq V\left( {HP_{n}^{0} } \right)\) and \(w \in V\left( {HP_{n}^{1} } \right)\).
By the definition of hyper Petersen networks, \(V\left( {HP_{n}^{0} } \right)\) and \(V\left( {HP_{n}^{1} } \right)\) interconnect via a perfect match. Hence, it must be either \(\left( {u,w} \right) \in E\left( {HP_{n} } \right)\) or \(\left( {v,w} \right) \in E\left( {HP_{n} } \right)\) which means the node \(w\) can only be adjacent to one of \(u\) and \(v\). Accordingly, \(u,v,w\) cannot be a cycle of length 3.
Consequently, \(HP_{n}\) has no cycle of length 3, and this lemma holds.
Lemma 3.3
Let \(HP_{n} \left( {n \ge 3} \right)\) be an \(n\)dimensional hyper Petersen network with faulty edges and \(min\{ deg\left( v \right)v \in V\left( {HP_{n} } \right)\} = r\) \(\left( {2 \le r \le n} \right)\). Let \(U\) be a node set of \(HP_{n}\), \(U \subseteq V\left( {HP_{n} } \right)\). Then \(\left {N_{{HP_{n} }} \left( U \right)} \right \ge r\) if \(1 \le \left U \right \le r\).
Proof

Case 1 \(\left U \right = 1\). In this case, there is one node \(v\) in \(U\).
According to \({ \hbox{min} }\{ { \deg }( v)v \in V\left( {HP_{n} } \right)\} = r\), \(\left {N_{{HP_{n} }} \left( v \right)} \right \ge r\) holds.

Case 2 \(\left U \right \ge 2\).
This case is further divided into two subcases. The first subcase is that no two nodes of \(U\) are adjacent to each other. The second subcase is that there are two nodes of \(U\) that are adjacent to each other.
 Case 2.1 No two nodes of \(U\) are adjacent to each other (see Fig. 3).
Let \(v_{1} \in U\) be a node of \(U\). Because no two nodes of \(U\) are adjacent to each other, then \(N_{{HP_{n} }} \left( {v_{1} } \right)\bigcap {U = \varPhi }\). According to \({ \hbox{min} }\{ { \deg }\left( v \right)v \in V\left( {HP_{n} } \right)\} = r\), \({ \deg }\left( {v_{1} } \right) \ge r\). Accordingly, \(N_{{HP_{n} }} \left( {v_{1} } \right) \subseteq N_{{HP_{n} }} \left( U \right)\) and \(\left {N_{{HP_{n} }} \left( U \right)} \right \ge r\) hold.
 Case 2.2 There are two nodes of \(U\) that are adjacent to each other (see Fig. 4).
Let \(v_{1} ,v_{2} \in U\) and \(\left( {v_{1} ,v_{2} } \right) \in E\left( {HP_{n} } \right)\). Since \({ \hbox{min} }\{ { \deg }\left( v \right)v \in V\left( {HP_{n} } \right)\} = r\), we suppose that \({ \deg }\left( {v_{1} } \right) = r + a\) and \({ \deg }\left( {v_{2} } \right) = r + b\left( {0 \le a,b \le n  r} \right)\). By Lemma 3.2, \(N_{{HP_{n} }} \left( {v_{1} } \right)\bigcap {N_{{HP_{n} }} \left( {v_{2} } \right) = \varPhi }\); hence, \(\left {N_{{HP_{n} }} \left( {v_{1} } \right)  v_{2} } \right = r + a  1\) and \(\left {N_{{HP_{n} }} \left( {v_{2} } \right)  v_{1} } \right = r + b  1\) hold. The following equation is obtained: \(\left {N_{{HP_{n} }} \left( {v_{1} ,v_{2} } \right)} \right = \left( {r + a  1} \right) + \left( {r + b  1} \right) = 2r + a + b  2\)
Since \(\left U \right \le r\) and \(2 \le r \le n\), \(\left U \right  2 \le r  2\) holds. Removing \(v_{1}\) and \(v_{2}\) from \(U\), there are \(\left U \right  2\) nodes in \(U  \left\{ {v_{1} ,v_{2} } \right\}\). Even if all the nodes of \(U  \left\{ {v_{1} ,v_{2} } \right\}\) are adjacent to \(v_{1}\) or \(v_{2}\), there are at least \(\left( {2r + a + b  2} \right)  \left( {r  2} \right) = r + a + b\) nodes belonging to \(N_{{HP_{n} }} \left( U \right)\). Accordingly, \(\left {N_{{HP_{n} }} \left( U \right)} \right \ge r + a + b \ge r\), and this lemma holds.
Theorem 3.2
Consider an \(n\)dimensional hyper Petersen network \(HP_{n} \left( {n \ge 3} \right)\) with faulty edges. Then, \(HP_{n} \left( {n \ge 3} \right)\) is \((r  1)_{i}\)diagnosable if \(min\{ deg\left( v \right)v \in V\left( {HP_{n} } \right)\} = r\) and \(2 \le r \le n\).
Proof
Let \(S_{1}\) and \(S_{2}\) be two node sets of the faulty \(HP_{n}\), \(\left {S_{1} } \right\), \(\left {S_{2} } \right \le r  1\), and \(S_{1} \bigcap {S_{2} = \varPhi }\). For \(S_{1}\), \(\left {N_{{HP_{n} }} \left( {S_{1} } \right)} \right \ge r\) by assumption and Lemma 3.2. Similarly, \(\left {N_{{HP_{n} }} \left( {S_{2} } \right)} \right \ge r\). Hence, \(S_{1}\) has at least one neighbor node that does not belong to \(S_{2}\), and \(S_{2}\) has at least one neighbor node that does not belong to \(S_{1}\). By Lemma 2.1, this theorem holds.
4 Intermittent Fault Diagnosability of \(HP_{n}\) Under the MM* Model
In this section, we investigate the intermittent fault diagnosability of \(HP_{n}\) under the MM* model.
Lemma 4.1
 1.
Suppose that \(u_{0} ,v_{0} \in R\), \(u_{1} ,u_{2} \in S_{1}\) and \(v_{1} ,v_{2} \in S_{2}\), then \((u_{1} ,u_{2} )_{{u_{0} }} \in C\) and \((v_{1} ,v_{2} )_{{v_{0} }} \in C\);
 2.
Suppose that \(u_{0} ,v_{0} ,u_{1} \in R\), \(u_{2} \in S_{1}\) and \(v_{1} ,v_{2} \in S_{2}\), then \((u_{1} ,u_{2} )_{{u_{0} }} \in C\) and \((v_{1} ,v_{2} )_{{v_{0} }} \in C\);
 3.
Suppose that \(u_{0} ,v_{0} ,v_{1} \in R\), \(u_{1} ,u_{2} \in S_{1}\) and \(v_{2} \in S_{2}\), then \((u_{1} ,u_{2} )_{{u_{0} }} \in C\) and \((v_{1} ,v_{2} )_{{v_{0} }} \in C\);
 4.
Suppose that \(u_{0} ,v_{0} ,u_{1} ,v_{1} \in R\), \(u_{2} \in S_{1}\) and \(v_{2} \in S_{2}\), then \((u_{1} ,u_{2} )_{{u_{0} }} \in C\) and \((v_{1} ,v_{2} )_{{v_{0} }} \in C\).
Lemma 4.2
Let \(u,v \in V\left( {HP_{n} } \right)\) and \(u \ne v\), then \(\left {N\left( {\left\{ {u,v} \right\}} \right)} \right \ge 2n  2\).
Proof

Case 1 When \(u\) and \(v\) belong to the same \(HP_{n  1}\), i.e., \(u,v \in V\left( {HP_{n  1}^{0} } \right)\) or \(u,v \in V\left( {HP_{n  1}^{1} } \right)\). According to the hypothesis that the lemma holds.

Case 2 Suppose that nodes \(u\) and \(v\) belong to \(HP_{n  1}^{0}\) and \(HP_{n  1}^{1}\), respectively. Without loss of generality, let \(u \in V\left( {HP_{n  1}^{0} } \right)\) and \(v \in V\left( {HP_{n  1}^{1} } \right)\). The rest discussion will be divided into two cases as follows.

Case 2.1 \(\left( {u,v} \right) \in E\left( {HP_{n} } \right)\). Because the node \(u\) has \(n  1\) neighbor nodes in \(HP_{n  1}^{0}\), the node \(v\) has \(n  1\) neighbor nodes in \(HP_{n  1}^{1}\), \(\left {N\left( {\left\{ {u,v} \right\}} \right)} \right \ge 2n  2\) holds by Definition 2.1.

Case 2.2 \(\left( {u,v} \right) \notin E\left( {HP_{n} } \right)\). Similar to Case 2.1, \(\left {N\left( u \right)\bigcap {HP_{n  1}^{0} } } \right = n  1\) and \(\left {N\left( v \right)\bigcap {HP_{n  1}^{1} } } \right = n  1\). According to the hypothesis, Nodes \(u\) and \(v\) are not adjacent, so each of them has a neighbor node in \(HP_{n  1}^{1}\) and \(HP_{n  1}^{0}\) by Definition 2.1, respectively, denoted as \(u^{\prime} \in V\left( {HP_{n  1}^{1} } \right)\) and \(v^{\prime} \in V\left( {HP_{n  1}^{0} } \right)\) respectively. According to whether \(u^{\prime}\) is a neighbor node of \(u\) in \(HP_{n  1}^{1}\), and whether \(v^{\prime}\) is a neighbor node of \(v\) in \(HP_{n  1}^{0}\), further discussion is as follows:

Case 2.2.1 \(u^{\prime} \in N_{{HP_{n  1}^{1} }} \left( u \right)\) and \(v^{\prime} \in N_{{HP_{n  1}^{0} }} \left( v \right)\), there are two common neighbor nodes between \(u\) and \(v\), then \(\left {N\left( {\left\{ {u,v} \right\}} \right)} \right \ge 2n  2\) holds.

Case 2.2.2 \(u^{\prime} \notin N_{{HP_{n  1}^{1} }} \left( u \right)\) and \(v^{\prime} \in N_{{HP_{n  1}^{0} }} \left( v \right)\), there is one common neighbor node between \(u\) and \(v\), then \(\left {N\left( {\left\{ {u,v} \right\}} \right)} \right \ge 2n  1\) holds.

Case 2.2.3 \(u^{\prime} \in N_{{HP_{n  1}^{1} }} \left( u \right)\) and \(v^{\prime} \notin N_{{HP_{n  1}^{0} }} \left( v \right)\), Similar to Case 2.2.2.

Case 2.2.4 \(u^{\prime} \notin N_{{HP_{n  1}^{1} }} \left( u \right)\) and \(v^{\prime} \notin N_{{HP_{n  1}^{0} }} \left( v \right)\), there is no common neighbor node between \(u\) and \(v\), then \(\left {N\left( {\left\{ {u,v} \right\}} \right)} \right \ge 2n\) holds.
Accordingly, this lemma holds.
Theorem 4.1
Let \(HP_{n}\) be an \(n\)dimensional hyper Petersen network, then \(HP_{n}\) is \((n  2)_{i}\)diagnosable under the MM* model.
Proof

Case 1 \(N\left( {u_{1} } \right)\bigcap {S_{1} = \phi }\). This means that all the neighbor nodes of \(u_{1}\) do not belong to \(S_{1}\). Because of \(\left {N\left( {u_{1} } \right)} \right = n\) and \(\left {S_{2} } \right \le n  2\), there must be at least one node \(u_{2}\) in \(N\left( {u_{1} } \right)\) so that \(u_{2} \in R\). According to the intersection relation between \(N\left( {u_{2} } \right)\) and \(S_{1}\), the proof will be divided into two subcases to be discussed separately.

Case 1.1 If \(\left {N\left( {u_{2} } \right)\bigcap {S_{1} } } \right \ge 2\), there is at least one node \(u_{3} \in N\left( {u_{2} } \right)\) so that \(u_{3} \ne u_{1}\) and \(u_{3} \in S_{1}\). Therefore, there is an A structure in \(HP_{n}\).

Case 1.2 If \(\left {N\left( {u_{2} } \right)\bigcap {S_{1} } } \right < 2\), it means that there is only one neighbor node \(u_{1} \in N\left( {u_{2} } \right)\) in \(S_{1}\), and the remaining \(n  1\) neighbor nodes do not belong to \(S_{1}\). According to \(\left {S_{2} } \right \le n  2\), there must be a node \(u_{3} \in N\left( {u_{2} } \right)\) so that \(u_{3} \in R\) holds. Therefore, there is a B structure in \(HP_{n}\).

Case 2 \(N(u_{1} )\bigcap {S_{1} \ne \phi }\). This means that there is at least one node \(u_{1}^{ '} \in S_{1}\). According to Lemma 4.2, the cardinality of \(N\left( {\left\{ {u_{1} ,u_{1}^{ '} } \right\}} \right)\) must be at least \(2n  2\). Therefore, \(\left {N\left( {\left\{ {u_{1} ,u_{1}^{ '} } \right\}} \right)\bigcup {\left\{ {u_{1} ,u_{1}^{ '} } \right\}  S_{1} } } \right \ge n + 2\). According to \(\left {S_{2} } \right \le n  2\), then \(\left {N\left( {\left\{ {u_{1} ,u_{1}^{'} } \right\}} \right)\bigcup {\left\{ {u_{1} ,u_{1}^{'} } \right\}}  S_{1} } \right > S_{2} \). Thus, there must be a node \(u_{2} \in N\left( {u_{1} } \right)\) or \(u_{2} \in N\left( {u_{1}^{ '} } \right)\), such that \(u_{2} \in R\). Without loss of generality, assume that \(u_{2} \in N\left( {u_{1} } \right)\). Similar to case1, we discuss separately in two subcases:

Case 2.1 If \(\left {N\left( {u_{2} } \right)\bigcap {S_{1} } } \right \ge 2\) then \(u_{2}\) has at least two neighbor nodes \(u_{1} \in S_{1}\) and \(u_{3} \in S_{1}\). Therefore, there is an A structure in \(HP_{n}\).

Case 2.2 If \(\left {N\left( {u_{2} } \right)\bigcap {S_{1} } } \right < 2\) then \(u_{2}\) has only one neighbor node \(u_{1} \in S_{1}\). Accordingly, \(\left {N\left( {u_{2} } \right)  S_{1}  S_{2} } \right \ge 1\) holds, which means that there is a neighbor node \(u_{3} \in N\left( {u_{2} } \right)\) and \(u_{3} \in R\). Therefore, there is a B structure in \(HP_{n}\).
If \(v_{1} \in S_{2}\), similar conclusions can be obtained. Therefore, for any two sets of nodes \(S_{1} \subset V\left( {HP_{n} } \right)\) and \(S_{2} \subset V\left( {HP_{n} } \right)\) in \(HP_{n}\), \(\left {S_{1} } \right \le n  2\), \(\left {S_{2} } \right \le n  2\), a combination satisfying one of the conditions in Lemma 4.1 can be obtained. Thus, this theorem holds.
5 Conclusions
The diagnosability of a multiprocessor system is an important factor in measuring its reliability. In a multiprocessor system, faults of processors can be categorized into three main types: permanent faults, intermittent faults, and transient faults. Because intermittent faults have more uncertainty, the diagnosis of intermittent faults is considered to be more difficult. Under the PMC model, this study shows that the intermittent fault diagnosability of an \(n\)dimensional hyper Petersen network is \(n  1\), and an \(n\)dimensional hyper Petersen network \(HP_{n} \left( {n \ge 3} \right)\) with faulty edges is \((r  1)_{i}\)diagnosable if \({ \hbox{min} }\{ { \deg }\left( v \right)v \in V\left( {HP_{n} } \right)\} = r\) and \(2 \le r \le n\). The intermittent fault diagnosability of an \(n\)dimensional hyper Petersen network is \(n  2\) under the MM* model. This work provides a theoretical basis for the practice of intermittent fault diagnosis and has a good application prospect.
Notes
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