Abstract

Diagnosability is an important metric parameter for measuring the reliability of multiprocessor systems. The pessimistic diagnosis strategy is a classic diagnostic model based on the PMC model. The class of folded Petersen cubes, denoted by , where and , is introduced as a competitive model of the hypercubes, which is constructed by iteratively applying the Cartesian product operation on the hypercube and the Petersen graph . In this paper, by exploring the structural properties of the folded Petersen cubes , we first prove that is diagnosable under the PMC model. Then, we completely derive that the pessimistic diagnosability of is under the PMC model. Furthermore, the diagnosability and the pessimistic diagnosability of the class of folded Petersen cubes, including the hypercube, folded Petersen graph, and hyper Petersen graph, are obtained.

1. Introduction

A multiprocessor system can be modeled as a graph, in which nodes (vertices) and edges correspond to processor and communication links, respectively. Throughout the paper, a graph and a system, a vertex and a processor, and an edge and a link are interchangeable.

The multiprocessor system has been increasingly adopted in the semiconductor technology, and the system reliability is crucial for multiprocessor systems. To maintain high reliability, multiprocessor systems should differentiate between fault-free processors and faulty ones. Determining all faulty processors is known as fault diagnosis. When all faulty processors can be evaluated precisely and is an upper bound of the number of faulty processors, we call the multiprocessor system as -diagnosable. The largest cardinality of the faulty set is named as the diagnosability of this system. Diagnosability of many famous networks had been studied; see [15] etc.

For the purpose of self-diagnosis of a system, several models have been proposed for diagnosing faulty processors in a multiprocessor system. Among the proposed models, the PMC model [6] was widely used. The PMC model allows each processor to perform diagnosis by testing the neighboring processors and observing their responses.

Observe that under a -diagnosable system, a node can only be tested by its neighbors. It is impossible to determine whether some processor is fault free or not when all the neighbors of are faulty. To improve the diagnosability, Kavianpour and Friedman [7] proposed the pessimistic diagnosis strategy, which is a classic strategy based on the PMC model. In this strategy, all faulty processors can be isolated within a set which has at most one fault-free processor.

Definition 1. Let be a system. is -diagnosable if all faulty processors can be isolated within a set of size at most such that at most, one fault-free processor is mistaken as a faulty one and the number of faulty processors is bounded by . The pessimistic diagnosability of is is - diagnosable}.

Using the PMC model with a pessimistic strategy, the pessimistic diagnosability has been receiving much attention for many well-known multiprocessor systems, such as hypercubes , Mbius cubes , enhanced hypercubes , -ary -cubes , alternating group graphs , hypercube-like network , star graph , and split-star networks ; see Table 1. More desired results can be found in [814] and the references therein.

Network topology is an important factor because it affects the performance of the network. Hypercubes [26] have been recognized as topologies of multiprocessor systems. The class of folded Petersen cubes, proposed by hring and Das [27], is constructed by iteratively applying the Cartesian product operation on hypercubes and the Petersen graph [2830]. For recent research about folded Petersen cubes, please refer to [3133] etc.

Although there are many results about diagnosability and the pessimistic diagnosability of many multiprocessor systems, little is known for folded Petersen cubes. In this paper, by exploring the structural properties of the folded Petersen cubes , we prove that is diagnosasable under the PMC model. Then, we completely determine the pessimistic diagnosability of under the PMC model. Furthermore, the diagnosability and the pessimistic diagnosability of the class of folded Petersen cubes, including the hypercube, folded Petersen graph, and hyper Petersen graph, are obtained.

2. Preliminaries

2.1. Terminologies and Notations

We provide Table 2 that contains most of the important notations used in this paper.

Let describe the link situation for a simple multiprocessor system. The processors in this system are denoted by an vertex set, and the links between each pair of processors are denoted by an edge set. Let be a processor. Denote by the set of processors which have a link to . For a set , define , named as the neighborhood of .

A graph is a subgraph of a graph if and . We say that a simple graph is regular when each processor has exactly neighbors. For any with , if is still connected, then, is -connected. The maximally connected subgraphs of a graph are its components. If a component has only one vertex, it is called trivial; otherwise, it is called nontrivial. Let be a vertex cut, the biggest component of is called a large component, and the remaining ones are called small components. Let is disconnected} be the connectivity of .

Suppose that and are two graphs with . Let be a perfect matching between the nodes of and . Then, is the graph with node set and edge set .

2.2. Folded Petersen Cube

Let and be two graphs. The Cartesian product of and , denoted by , is the graph with node set , and the vertices and are adjacent if and only if and or and . Under isomorphism, the operator is associative and commutative. For any graph and any positive integer , we define and if . The hypercube (where ) is defined as and . Thus, we also write .

The Petersen graph was introduced by Chartrand and Wilson [28]; see Figure 1. Obviously, the Petersen graph has an outer cycle, an inner cycle, and five spokes joining them. For , represents the -dimensional folded Petersen graph and (where ) represents the hyper Petersen graph [34].

For any , denote by the subgraph of induced by the node set . Thus, is recursively constructed from for . For any two nodes and of , suppose that and . If , then, and are adjacent if and only if and are adjacent in ; otherwise, and are adjacent if and only if is adjacent to in and .

From the construction of folded Petersen graphs, it is obvious that any node of has neighbors in and other three neighbors (called extra neighbors) in , where and are adjacent in .

Lemma 2 (see [27]). The folded Petersen graph is a node and edge transitive regular graph of degree and of node connectivity . For a node , the three extra neighbors of are in distinct , where and are adjacent in . Furthermore, for any two nodes , , where .
The class of folded Petersen cubes, where , and , was introduced as a competitive model of the hypercubes. In particular, and . Clearly, is a triangle-free -regular graph with vertices.

Lemma 3 (see [27]). The folded Petersen cube is a regular graph with degree and connectivity . Furthermore, can be viewed as , where is a perfect matching between two ’s.

2.3. PMC Model

In self-diagnosable systems, there are several methods which had been introduced to diagnose faulty processors. The PMC model [6] allows each processor to perform diagnosis by testing the neighboring processors and observing their responses. In the PMC model, a test syndrome collects all test results. Let . is said to be compatible with a syndrome if can be produced from the condition that all nodes in are faulty and all nodes in are fault free. Let is compatible with . Two distinct sets are indistinguishable if and distinguishable otherwise. The symmetric difference of two sets and is .

Dahbura and Masson [35] proposed a characterization for a pair of sets to be distinguishable under the PMC model.

Lemma 4 (see [35]). Let be a graph. For any two distinct sets , is a distinguishable pair under the PMC model if and only if there exist two nodes and satisfying ; see Figure 2.

3. Main Results

In this section, we first study the structure properties of . Using the structural properties and some basic lemmas, we can obtained the diagnosability and the pessimistic diagnosability of the folded Petersen cube network .

The following inequalities are useful for our proof.

Lemma 5. Let and be non-negative integers with , and . The following inequalities hold.

Proof. Since the proof for the three statements are similar, we just prove (1), and the proof for (2) and (3) are left for readers.

The following is the proof by induction on .

The initial step is as follows: if , then, holds for any integer . If , then, holds for any integer .

The induction step is as follows: assume that and the statement holds for , i.e., . Then,

Hence, the statement holds for as well.

3.1. Structure Properties of

Lemma 6 (see [16]). Let be a connected graph and . If , then, ; otherwise, .

Lemma 7. If and are two distinct nodes in with , , and , then, .

Proof. If and are adjacent, then, by Lemma 3; otherwise, since has no triangles, any two nonadjacent nodes have at most one common neighbor, and therefore, .

Lemma 8. Let be a subset of for . If , then, .

Proof. The lemma is proved by the induction on . If , then, ; the result holds by Lemma 7. Assume that the lemma is true for , where is an integer with . In the following, we consider . Recall that is constructed by disjoint s, denoted by for . Let and for . W.l.o.g., we may assume that . There are the following cases.

Case 1. .

In this case, for all , . Clearly, because of . If , then, by Lemma 7, we know the result holds. Now, assume that and for . By Lemma 2, . Since is regular and is isomorphic to , we have for .

Case 2. .

By inductive hypothesis in , . We distinguish the following two cases.

Case 2.1. .

By Lemma 2, we know

Case 2.2. .

There exists a such that . By Lemma 2,

If , then, . Note that and are node disjoints; we have for .

Now, consider . By induction hypothesis in ,

Thus, for .

Case 3. .

Since and

For , we have by Lemma 6 and by Lemma 2. We distinguish the following two cases.

Case 3.1. .

By Lemma 2, we know for .

Case 3.2. .

There exists such that . Since and , we have . Recall that . If , then,

If , then, for , and therefore, by Lemma 6. Hence, we have for .

Lemma 9. For with and , it holds that .

Proof. Proof by induction on .
If , then and the result holds by Lemma 7. Assume that the lemma is true for with . Now, we consider as follows. Recall that is constructed by two disjoint s, denoted by and . For , let . W.l.o.g., we may assume that . Since is regular and triangle free, if , then, . Next, let . It follows that . We distinguish the following two cases.

Case 1. .

By the induction hypothesis in , .

If , thenm .Thus,

If , then, .Thus,

It remains to assume that . Then, . Thus,

Case 2. .

Since is and for , Lemma 6 implies that .

If , then,

If , then, . Thus,

Therefore, the lemma is true for as well.

Lemma 10. For with , , , and , it holds that .

Proof. Proof by induction on .

If , then, and ; the lemma holds by Lemma 8.

Assume and the result holds for . We consider the result for as follows.

If , then, , the lemma holds by Lemma 9. So, let .

By Lemma 3, we know that can be viewed as , where is a perfect matching between two ’s. In other words, contains two copies of , denoted by and , respectively. For , let . W.l.o.g., we may assume that .

If , the result holds by Lemma 7. Next, let . It implies that . We distinguish the following two cases.

Case 1. Let .

By the induction hypothesis in ,

If , then, . It leads to

If , then .Thus,

It remains to assume that . Then,

Thus,

Case 2. Let .

In this case, we have where the inequality follows from Lemma 5 (1). By Lemma 6, .

If , then,

If , then, . Thus,

3.2. T-Diagnosability of

In what follows, we will discuss the diagnosability of under the PMC model.

Lemma 11. Let be the folded Petersen cube for , and . Then, under the PMC model.

Proof. Let be any vertex in . Let and . Assume that both and are faulty sets. Obviously, , , , and . If and are distinguishable under the PMC model, then, there exists some vertex such that , a contradiction. Thus, and are indistinguishable under the PMC model. It leads to .

Lemma 12. Let be the folded Petersen cube for , and . Then, under the PMC model.

Proof. Let and be two distinct faulty sets such that . Since and are distinct, , which implies that or . W.l.o.g., let . To complete the proof, it is sufficient to show that and are distinguishable under the PMC model.
Suppose to the contrary that and are indistinguishable. If , then, . Since , we have , a contradiction to the assumption. Thus, . Since the assumption that and are indistinguishable under the PMC model, there exist no edges between and . Note that is connected. So, is a vertex cut of and . Recall that , i.e., . Thus, a contradiction.

The following theorem follows directly from the previous two lemmas.

Theorem 13. Let be the folded Petersen cube for , and . Then, under the PMC model.

The following corollaries are a straightforward consequence of Theorem 13.

Corollary 14. Let be the -dimensional hypercube. Then, under the PMC model.

Corollary 15. Let be the -dimensional folded Petersen graph. Then, under the PMC model.

Corollary 16. Let (where ) be the hyper Petersen graph. Then, under the PMC model.

3.3. Pessimistic Diagnosability of

Chwa and Hakimi [36] derived a characterization for a graph to be diagnosability.

Lemma 17 (see [36]). Let be the representation of a system. is diagnosable if and only if for each integer with and each with ; it holds that .
Tsai and Chen [37] further derived another characterization for a graph to be diagnosability.

Lemma 18 (see [37]). A graph is diagnosable if and only if for each set with ; the graph has at most one trivial component, and each nontrivial component of satisfies .

Lemma 19. Let be the folded Petersen cube for , , and . Then, for each with and , it holds that .

Proof. We consider the following two cases according to the value of .

Case 1. Let .

By the assumption, . Moreover, and by Lemma 5 (3). Hence, we can deduce that

By Lemma 6, .

Case 2. Let .

By the assumption, . By Lemma 8, .

Theorem 20. Let be the folded Petersen cube for , and . Then, .

Proof. By Lemmas 17 and 19, . It remains to show that for , and .

Suppose to the contrary that . Let be an edge in and let . Since has no triangles, . The subgraph induced by is a connected component of . By Lemma 18,

a contradiction.

The following corollaries are obvious from Theorem 20.

Corollary 21. Let be the -dimensional hypercube. Then, .

Corollary 22. Let be the -dimensional folded Petersen graph. Then, .

Corollary 23. Let (where ) be the hyper Petersen graph. Then, .

4. Concluding Remarks

In this paper, by exploring structural properties of the folded Petersen cubes , we prove that is diagnosasable under the PMC model. Moreover, we study the pessimistic diagnosability of folded Petersen cubes and obtain for , and . As corollaries, the diagnosability and the pessimistic diagnosability of hypercube, folded Petersen graph, and the hyper Petersen graph are obtained. Another direction of our study in this paper is to investigate the conditional diagnosability of these graphs.

Data Availability

No data were used to support this study.

Disclosure

This work was accomplished while the first author was visiting Zhejiang Normal University.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author Yingli Kang is supported by NSFC (11901258) and ZJNSF (LY22A010016). The second author Shuai Ye is supported by the General Project of Zhejiang Provincial Education Department (Y202146807).