The Strong Local Diagnosability of a Hypercube Network with Missing Edges

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 n dimensions, denoted by Qn, the SLD of a node is equal to its degree when n ⩾ 4. 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 n − 3 . 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 n-dimensions is greater than 3, then the SLD of any node is still equal to its RD provided that the number of faulty edges does not exceed 7 n − 3 − 1. If the RD of each node is greater than 4, then the SLD of each node is also equal to its RD, no matter how many faulty edges exist in Qn.


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][2][3][4][5][6][7][8][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 n-dimensional hypercube network, denoted as Q n , consists of 2 n nodes and n2 n−1 edges.A binary n-bit string, e.g., x 1 x 2 , … , x n , can be employed to denote the address of a node x in Q n .If the address of node u and the address of node v are different in exactly one bit position, then they are connected by a link, and vice versa.For two nodes u and v in Q n , if x and y are connected by a link and their addresses differ in the ith position, then this edge is called the ith-dimensional edge.Q n can be decomposed in two n − 1 -dimensional hypercube networks Q 0 n−1 and Q 1 n−1 along the ith-dimensional edges.Let v 0 ∈ V Q 0 n−1 .Then there must be another node We call this edge the crossing edge between Q 0 n−1 and Q 1 n−1 .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 t-diagnosable system and of a sequentially t-diagnosable system.Since then, results corresponding to the PMC model have been widely reported (see [9,11,[13][14][15][16][17][18][19]).In the PMC model, when we consider the diagnosability of a system S 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 S with missing edges depends on the upper limit of the minimum degree of S. In practice, the probability that all neighbors of a node are synchronously faulty is very low.For example, consider an n-dimensional hypercube network Q n ; the number of node subsets such that its cardinality is n is C n 2 n .However, the number of node subsets such that its size is n and it incorporates all neighbors of some node is at most 2 n .The ratio 2 n /C n 2 n 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 tdiagnosable system (STDS).Lai et al. [14] claimed that a STDS is a t + 1 -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 t + 1 -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 S denote the integrating system produced by Q r and Q s , where s > >r.The diagnosability of S may be less than r.However, the system may diagnose all faulty processors even if their number is greater than r and less than s.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 Q n is n, where n ⩾ 4.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 Q n is studied.Finally, in Section 5, we discuss the conclusions of our paper.

Preliminaries
A system is usually described by a graph H V, E , where E represents all communication edges between processors and V 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 u in H V, E , a node v is said to be its neighbor if In the PMC model, if x, y ∈ E, then x and y can be used to test each other.We use the order pair x, y to represent that x tests y.In this situation, x is the tester, and y is the tested node.The PMC model assumes that if y is diagnosed to be faulty (fault-free) by x, then the outcome of x testing y is 1 (0), denoted by σ x, y = 1 (σ x, y = 0).We use σ = σ x, y | x, y ∈ E to denote a syndrome.The collection of all faulty nodes in H is called a faulty set.It is possible that any subset of V 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 H, the cardinality of a maximum faulty node set that can be identified by H is called its diagnosability, denoted as t H .Given a syndrome σ, a set S ⊆ V H is called consistent with σ if the following condition is true: if ∀x ∈ V H − S and x, y ∈ E, then σ x, y = 1 if and only if y ∈ S.
We use σ S to denote the set of syndromes produced by faulty node set S. Two subsets X, Y ⊆ V H are called distinguishable if σ X ∩ σ Y = ∅.If X and Y are distinguishable, then X, Y is said to be a distinguishable pair; otherwise, X, Y is said to be an indistinguishable pair.Let XΔY = X − Y ∪ Y − X .In addition, we need some previous results concerning the t-diagnosable  1).
The following lemmas are related to the concept of node diagnosability: Lemma 3 (see [20]).Suppose that H V, E is a system, z ∈ V. H is locally t-diagnosable at z if and only if for each subset X ⊂ V with |X| = l, l ∈ 0, 1, ⋯, t − 1 , and z ∉ X, the number of nodes in the connected component containing z in H − X is greater than 2 t − l .
Lemma 4 (see [20]).Suppose that H V, E is a graph, v ∈ V with deg v = k.Then, the locally diagnosability (LD) of node v does not exceed k.
Lemma 6 (see [20]).A t-diagnosable system H is locally t-diagnosable at each node.Conversely, if H is locally t-diagnosable at each node, then H is t-diagnosable.
By Lemma 1 and Lemma 5, we conclude that a tdiagnosable system must be t-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 t-diagnosable system follows [14].
The strong diagnosability (SD) of a system is the maximum number value of t such that the system is a strong t-diagnosable system.

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 t-diagnosable system but may not be a t + 1 -diagnosable system [13].Since the probability that all neighbors of some node are faulty simultaneously is rather small, a STDS is almost t + 1-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 Definition 3. Let H V, E be the diagnostic graph of a system H and x ∈ V. H is called strong locally t-diagnosable at node x if it is locally t-diagnosable at node x and the following condition is true.
For any two different subsets A system H is said to be strong locally t-diagnosable if for each node x in H, H is strong locally t-diagnosable at node x.
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 tdiagnosable system and a strongly t-diagnosable system: Since X ≠ Y, there exists a node y such that y ∈ XΔY and N y ⊄X ∩ Y. Hence, H is not strong locally t-diagnosable at node y, a contradiction to the postulation that H is strong locally t-diagnosable.This completes the proof.
The above proposition proposes a sufficient but unnecessary condition to test if the system S is strongly t-diagnosable.Next, a necessary and sufficient condition for testing the strong t-diagnosability of a triangle-free system is presented.Proposition 3. Let H V, E be a triangle-free system and t ⩾ 3 be an integer.If H is strongly t-diagnosable, then H is strong locally t-diagnosable, and vice versa.Proof 3. By the proof of Proposition 2, the sufficiency holds.
Necessity 1.Let H be a strongly t-diagnosable system; then, H is t-diagnosable.Let x ∈ V.By Lemma 6, we have that H is locally t-diagnosable at vertex x.Next, by Definition 3, we need to prove only that for any two different subsets X, Y ⊂ V with |X| ⩽ t + 1 and |Y| ⩽ t + 1, if X, Y is an indistinguishable pair, then H has at least a node x ∈ XΔY and N x ⊆ X ∩ Y.In contrast, assuming that N x ⊈ X ∩ Y, we derive a contradiction.By Definition 2, there is at least one node Here, we need to consider only the situation where x ∈ X − Y, as the proof of the situation in which x ∈ Y − X can be similarly obtained.In the rest of this proof, four situations are considered according to the subjections of node y: (1) By the hypothesis, we have that X − Y = x and Y − X ⩽ 1.We first discuss the situation where Y − X = 1.Suppose that Y − X = z .By the proof of Case 2, we have that x is adjacent to z and that there is no edge between z and any node in Thus, the degree of z is no greater than 2, a contradiction to deg z ⩾ t ⩾ 3 (Lemma 4).Second, we consider the situation where Y − X = 0. Since there is no edge between x and any node in V − X ∪ Y , N x ⊆ X ∪ Y; thus, N x ⊆ X, and therefore, N x ∪ x ⊂ X.On the other hand, since N x ⩾ t, X = x ∪ N x , and hence, Y = N x , 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 H V, E , the strong diagnosability of H is equal to min the SLD at node r | r ∈ V .
The following sufficient and necessary condition can be applied.Theorem 2. In a graph H V, E , x ∈ V. Thus, H V, E is strong locally t-diagnosable at node x if and only if, for any S ⊂ V, |S| = l, l ∈ 0, 1, ⋯, t , and x ∉ S, the following conditions hold: Next, we consider the situation l = t.From last paragraph, we have that H is locally t-diagnosable at node x.By Definition 3, we only need to prove that for any two different subsets and x ∈ XΔY.By the assumption that H V, E is strong locally t-diagnosable at node x, we have that H V, E is locally t-diagnosable at node x.Moreover, we have that X, Y is an indistinguishable pair.By Definition 3, we conclude that N x ⊆ X ∩ Y = S.This completes the proof of necessity.
For any S ⊂ V, |S| = l, l ∈ 0, 1, ⋯, t , and x ∉ S. We will prove that the two conditions in Theorem 2 hold.Let Consider the following cases: Next, we present the type I structure, which is used to verify the SLD of a given node.Definition 4. Suppose that H V, E denotes a diagnostic graph of a system and that x is a node in V.The type I structure of the node x can be decomposed in three node sets (L 1 , L 2 , and L 3 ) and two edge sets (E 1 , E 2 ), as shown in Figure 2.These sets are defined as follows:

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. Q n is strong locally n-diagnosable at each node, where n ⩾ 4.
Proof 6.Let v ∈ V Q n be an arbitrary node.For each S ⊂ V Q n , |S| = l, l ∈ 0, 1, ⋯, n , and v ∉ S, we show that the two conditions of Theorem 2 are both true.
Case 9 l ≠ n .Noting that the connectivity of According to Theorem 3, the SLD of every node of Q n is the same as its degree, with n ⩾ 4. 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 E ′ ⊂ E to denote an edge subset and Q n ′ to denote an incomplete hypercube network of n dimensions created by removing E′ from the hypercube network Q n .In the following, we show that the SLD of every node is the same as its RD in Q n ′ , even if the cardinality of E ′ can reach n − 3 .We now give an example to explain that the result is not true if the cardinality of E ′ is n − 2 .In Figure 3, Then, the SLD of every node of Q n ′ is exactly the same as its RD.Next, we show that when 0 5, we conclude that the only case stopping the condition from being satisfied is that in which Consider the following two cases: Next, we prove that the two conditions of Theorem 2 hold.For the sake of convenience, let C stand for the connected component containing Case 15.First, we consider the condition that l ⩽ k.We will prove that |V C | ⩾ 2 k + 1 + 1 − l + 1.Consider the following cases: x, y has at least one node, e.g., x, which is not v.Let by inductive hypothesis and Theorem 2, we have that , we can choose two distinct nodes u and w such that either both u and w are neighbors of v or u is a neighbor of v and w is a neighbor of u.Suppose that u, u′ and w, w′ are two crossing edges in Therefore, v′ has two neighbors, e.g., x and y, in Q 1 n−1 − E 1 ′.Consider the following cases:

Complexity
Case 21 0 ⩽ l ⩽ k − 1 .By inductive hypothesis and Theorem 2, we have that Hence, by inductive hypothesis and Theorem 2, we have that Case 23.Next, we consider the condition that l = k + 1. Suppose that N Q n ′ v ⊆ S. Consider the following cases.In summary, Q n − E′ is strong locally k + 1 -diagnosable at the node v.
Clearly, the following corollary follows Theorem 4; hence, its proof is omitted.
In the previous section, we concluded that the SLD of a node is its RD in an incomplete hypercube network Q n ′ created by removing the edge set E′ ⊂ E Q n with |E′| ⩽ n − 3 from Q n .In the following, we discuss the problem of the upper of |E ′ | such that the following two conditions are satisfied: (1) the RD of each node in Q n − E ′ , denoted by Q n ′ , is at least 4; (2) the SLD of every node Q n ′ is the same as its RD.Now, we consider an example in which the SLD of a node in Q n ′ differs from its RD in Q n ′ with |E ′ | = 7 n − 3 , even though the minimum degree of nodes in Q n ′ is more than 3. Assume that x, x i ∈ V, 1 ⩽ i ⩽ 7, the adjacent situations of which are shown in Figure 6.Let Proof 8. Before proving this theorem, we define some notions for further discussion.Consider another structure of a node v obtained by extending its type I structure, called the type I-EX structure of node v, in the following way: insert a node set L 4 and an edge set E 3 in the type I structure of node v, where After removing E′ from Q n , the type I-EX structure of node v becomes another new structure, which implies that Condition 1 of Theorem 2 holds, we consider the following cases: then there exists some node v 0 ∈ L 3 ′ such that there is exactly one node in L 3 ′ that can connect to v 0 .Then, L 4 ′ has at least one node, e.g., v 0 ′ , such that v 0 , v 0 ′ ∈ E ′ .In other words, V C ⩾ 9, which contradicts the assumption that which implies that there exist 7 nodes in Q n ′ whose RDs are all 3.This contradicts the assumption that 0 Finally, we discuss the situation where the RD of each node in Q n − E ′ exceeds 3.
By Theorem 6, we have the following corollary.
locally r-diagnosable at each node.

Conclusions
For a system, if we consider only its global properties and ignore its local status, we may lose some interesting and important information.This paper proposes the concept of SLD and derives some conditions for testing the SLD of a node in a system.Moreover, we discuss the relationship between the SLD and the SD of a system.Based on this 9 Complexity relationship, we obtain the SD of a system by determining the SLD of every node in the system.Our results show that the SLD of a node in Q n is n, where n ⩾ 4.Then, we consider the SLD of an incomplete hypercube network Q n − E ′ , where E′ ⊂ E Q n .When 0 ⩽ |E′| ⩽ n − 3, the SLD of a node is the same as its RD in Q n − E′.The above mentioned result still holds in the following two cases: It is worth mentioning that haptic identification [25] is an important method of identification, and it is a natural idea to combine haptic identification with the strong local fault identification mentioned by the paper.Our future research work is to explore the strong local haptic identification of a system.

Figure 1 :
Figure 1: An example of a distinguishable pair X, Y represent the number of nodes (edges) in the type I structure.

Proof 7 .
We prove this theorem by induction n.First, let us consider the situation where n = 4; in this case, |E ′ | ⩽ 1.For |E ′ | = 0, based on Theorem 3, the result holds.Assume that |E ′ | = 1.We consider the type I structure of node v in Q 4 , as shown in Figure 4.It is clear that |L 1 | + |L 2 | + |L 3 | = 1 + 4 + 6 = 11.If E′ ⊂ E 1 , then the RD of v is 3.We can use a similar argument to that used in the proof of Theorem 3 to show that the SLD of node v is 3.If E ′ ⊂ E 2 , then the RD of v is 4. Let S ⊂ V Q 4 ′ and v ∈ S, with |S| = l and 0 ⩽ l ⩽ 4. When l = 4 and N v ⊆ S, Condition 2 of Theorem 2 holds.When l = 4 and N v ⊆ S, consider the following cases.Case 11 |N v − S| ⩾ 2 .Then the number of nodes of the connected component containing v in Q 4 ′ − S is at least 3, Condition 2 of Theorem 2 holds.Case 12 |N v − S| = 1 .Then |N v ∩ S| = 3.Let u ∈ N v − S, then N u ∩ L 3 has at least 2 nodes, which implies |N u ∩ L 3 − S| ⩾ 1.So, the number of nodes of connected component containing v in Q 4 ′ − S is at least 3.In other words, Condition 2 of Theorem 2 holds.

Figure 4 :
Figure 4: An illustration of the difference between the strong locally diagnosability and RD for a node in Q n ′.

Figure 5 :
Figure 5: Type I structure of the node v in Q 4 .
then there is at least one node in N Q n −E′ , e.g., z, such that z ∉ S.Then, x, y, z belong to a connected component in Q n − E ′ − S. Condition 2 of Theorem 2 is satisfied.

Figure 6 :
Figure 6: Illustration of t sl Q n − E′ x being equal to its RD.

Corollary 3 .
the type II structure of node v. On the other hand, the assumption that for eachv ∈ V Q n ′ , deg Q n ′ v ⩾ 3 implies that |L 3 ′ | = 3, where there exists at most one node in L 3 ′ that is shared by the two nodes in L 2 ′ as their common neighbor.Furthermore, |L 4 ′ | ⩾ 1 can be obtained, which implies that V C ⩾ 7.This is a contradiction.Case 29 h − l = 1 .Note that 2 h + 1 − l + 1 = 5 and 1 + 2 h − l + h − l/3 = 4; hence, it is sufficient that V C ≠ 4. Suppose that V C = 4. Since h − l = 1, it follows that |L 1 ′ | = 1 and |L 2 ′ | = 1 in the Type II structure of node v.The assumption that for each v ∈ V Q n ′ , deg Q n ′ v ⩾ 3 implies that |L 3 ′| ⩾ 2, which results in |L 4 ′| ⩾ 1.Hence, V C ⩾ 5, which contradicts the assumption that V C = 4.By Theorem 5, we can obtain the following corollary.In Q n (n ⩾ 4), suppose that E ′ ⊂ E Q n , with 0 ⩽ |E ′ | ⩽ 7 n − 3 − 1.Then, Q n − E ′ is strongly r-diagnos-able, where r represents the minimum degree of all nodes in Q n − E′ and r ⩾ 3.
X, Y is a distinguishable pair, which is a contradiction.Thus, |S ′ | = l = t, which implies that |XΔY| ⩽ 2. By Condition 2, we have that the number of nodes of the connected component containing x in H − S ′ exceeds 2. Hence, H has two nodes y ∈ V − X ∪ Y and z ∈ XΔY such that y, z ∈ E. Hence, X, Y is a distinguishable pair, which is a contradiction.
then the number of nodes of the connected component containing x in H − S exceeds 2 t + 1 − l .Condition 2 If l = t, then either (a) the number of nodes of the connected component containing x in H − S exceeds 2 or (b) N x ⊆ S. Proof 4 (sufficiency).Let C denote the connected component containing x in H − S. By Condition 1, for l ≠ t, we have that 2 t − l + 1 < 2 t + 1 − l + 1 ⩽ |V C |.According to Lemma 3, H is locally t-diagnosable at node x.Next, we need to prove that for any two different subsets X, Y ⊂ V with |X| ⩽ t + 1, |Y| ⩽ t + 1 and x ∈ XΔY, if X, Y is an indistinguishable pair, then N x ⊆ X ∩ Y. Suppose that there exist two different subsets X, Y ⊂ V with |X| ⩽ t + 1, |Y| ⩽ t + 1 and x ∈ XΔY such that X, Y is an indistinguishable pair and N x ⊈ X ∩ Y. Let S ′ = X ∩ Y; then, |S ′ | = l ⩽ t.If l ≠ t, by Condition 1, the number of nodes of the connected component containing x in H − S ′ is greater than 2 t + 1 − l .Moreover, the number of nodes of the connected component containing x in H − S ′ is greater than |XΔY|, which implies that H has two nodes y ∈ V − X ∪ Y and z ∈ XΔY such that y, z ∈ E.
and then we have that |XΔY| ⩽ 2 t + 1 − r .Moreover, by Condition 1, we have that r = t, which implies that |XΔY| ⩽ 2. Since X, Y is an indistinguishable pair, there is no edge from V − X ∪ Y to XΔY.So, the number of nodes of the connected component containing x is no more than |XΔY| ⩽ 2.Moreover, N x ⊆ S′ = X ∩ Y, a contradiction.Necessity 2. We consider Condition 1 first.Suppose that for some It is clear that |X| ⩽ t + 1, |Y| ⩽ t + 1 and x ∈ XΔY.Since C is a connected component of H − S, H does not have an edge y, z ∈ E such that y ∈ XΔY and z ∈ V − X ∪ Y .By Lemma 2, X, Y is an indistinguishable pair.Since H is strong locally t-diagnosable at node x, N x ⊆ S. Hence, |N x | ⩽ |S| ⩽ l ⩽ t − 1.On the other hand, the assumption that H is strong locally tdiagnosable at node x implies that H is t-diagnosable at node x.By Lemma 4, we have that |N x | ⩾ t, which is a contradiction.Hence, Condition 1 is necessary.Now, let us consider Condition 2. Suppose that |S| = l = t.Let L denote the connected component containing x in H − S. If the number of nodes in L is less than or equal to 2, then V L can be decomposed in two subsets R 1 and R 2 satisfying the following conditions: R 1 is strong locally t-diagnosable at y, the number of nodes of the connected component containing y in H ′ − S ′ exceeds 2 t + 1 − l ′ , which implies the number of nodes of the con-nected component containing y in H − S exceeds 2 t + 1 − l ′ .Moreover, the number of nodes of the connected component containing y in H − S exceeds 2 t + 1 − l .So, Condition 1 of Theorem 2 holds.
Case 7 l ′ ≠ t .An argument being similar to Case 5 can be used to obtain the following result: the number of nodes of the connected component containing y in H ′ − S ′ exceeds 2 t + 1 − l′ , which implies that the number of nodes of the connected component containing y in H − S exceeds 2. So, Condition 2 of Theorem 2 holds.Case 8 l′ = t .Then S′ = S. Since H′ V′, E′ is strong locally t-diagnosable at y and N H y ⊆ V′, then either (a) the number of nodes of the connected component containing y in H′ − S′ exceeds 2, which implies the number of nodes of the connected component containing y in H − S exceeds 2, or (b) N H ′ y ⊆ S′, which implies N H y ⊆ S. So, Condition 2 of Theorem 2 holds.

8
Complexitycalled the type II structure of node v.The set of nodes L 1 (L 2 ; L 3 ; L 4 ) becomes another new set of nodes L 1 ′ (L 2 ′ ; L 3 ′ ; L 4 ′ ) in the type II structure of node v. Similarly, the set of edges E 1 (E 2 ; E 3 ) becomes another new set of edgesE 1 ′ (E 2 ′ ; E 3 ′ ) in the type II structure of node v. Let deg Q n −E ′ v = h and S ⊂ V Q n with |S| = l and 0 ⩽ l ⩽ h.By the type II structure of node v, we have that the number of nodes of connected component containing v in Q n ′ − S is smallest if and only if S ⊂ L 2 ′.So, we need to consider only the case of S ⊂ L 2 .If |S| = h, it is obtained that N v ⊆ S, which satisfies Condition 2 of Theorem 2. Now, we consider the case of 0 ⩽ |S| ⩽ h − 1.For the sake of convenience, we introduce another new structure of node v, called type III structure of node v, which is obtained by removing S from the type II structure of node v.The set of nodes L 1