Diagnosability of star graphs with missing edges

Diagnosability of star graphs with missing edges

q

Chieh-Feng Chiang

a

, Guo-Huang Hsu

b

, Lun-Min Shih

c

, Jimmy J.M. Tan

a,⇑

a_{Department of Computer Science, National Chiao Tung University, Hsinchu 30050, Taiwan} b

Industrial Technology Research Institute, Hsinchu City, Taiwan

c

Department of Continuing and Extension Education, Da Yeh University, Taiwan

a r t i c l e

i n f o

Article history:

Received 25 August 2008

Received in revised form 27 July 2011 Accepted 7 November 2011 Available online 16 November 2011 Keywords:

Star graph

Comparison diagnosis model MM⁄

diagnosis model Local diagnosability Extended star structure

Strong local diagnosability property

a b s t r a c t

In this paper, we study the system diagnosis on an n-dimensional star under the compar-ison model. Following the concept of local diagnosability[3], the strong local diagnosability property[7]is discussed; this property describes the equivalence of the local diagnosability of a node and its degree. We prove that an n-dimensional star has this property, and it keeps this strong property even if there exist n  3 missing edges in it.

 2011 Elsevier Inc. All rights reserved.

1. Introduction

In recent years, with the continuing advancements in semiconductor technology, large multiprocessor systems such as very-large-scale integration (VLSI) systems have become increasingly popular. Such systems must be capable of uninter-rupted processing, and therefore, the reliability of the processors in these systems should be considered. The diagnosis of such systems involves the identification of all the faulty processors in the system. The diagnosability of the system refers to the maximum number of faulty processors that can definitely be identified.

Several approaches to system diagnosis have been developed in previous researches. One major approach, called the com-parison diagnosis model, was proposed by Maeng and Malek[13,14]. In this model, diagnosis is performed by simulta-neously sending two identical signals from a processor to two other linked processors and then comparing the responses. The test results are collected and analyzed to identify all the faulty processors. Following the traditional concept of diagnos-ability, many variants of diagnosability measurements have been presented. A different measurement called conditional diagnosability was proposed[8], and a more precise concept called strong diagnosability has been widely applied to various networks[4–6].

In contrast to the traditional concept of diagnosability, Chiang and Tan[3]introduced a different concept for system diag-nosis called local diagnosability; this method requires only the correct identification of the status of a single processor. Each processor has its own local diagnosability, and there exists a strong relationship between the local diagnosability and the traditional diagnosability. In the comparison diagnosis model for a given processor, a local structure called an extended star has also been presented for guaranteeing a processor’s local diagnosability.

0020-0255/$ - see front matter  2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.11.012

q

This work was supported in part by the National Science Council of the Republic of China under Contract NSC 99-2221-E-009-083-MY3.

⇑Corresponding author. Tel.: +886 3 5715900; fax: +886 3 5721490. E-mail address:jmtan@cs.nctu.edu.tw(J.J.M. Tan).

Contents lists available atSciVerse ScienceDirect

Information Sciences

Among all well-known topologies, the star graph is one of the most popular ones. Its features include node symmetry, edge symmetry, regular and low degree of node, and small diameter. Since its introduction, this topology has attracted con-siderable attention. Some studies have discussed the diameter and fault diameters[9,15,16]. When linearly many vertices are deleted in a star graph, the resulting graph has a large connected component containing almost all remaining vertices

[2]. The problem of embedding a linear array of vertices (or a ring) into the star graph has also been solved, even when there exist some vertex faults in the target star graph[10]. With edge faults, the star graph has been proved to have fault-tolerant Hamiltonian laceability[12]. The robustness of star graphs with edge faults has been addressed[11], and the improvement of bounds on edge failure tolerance has also been investigated[18]. For system diagnosis, Zheng et al.[20]showed that the traditional diagnosability of an n-dimensional star is n  1 for n P 4.

In this paper, we study system diagnosis by following the concept of local diagnosability[3]. Based on this concept, we obtain a simple proof of the fact that the diagnosability of an n-dimensional star Snis n  1 for n P 4; this is the same result

as that obtained by Zheng et al.[20]. Moreover, we study the diagnosability of a star graph in the presence of arbitrary dis-tributed missing edges under the comparison diagnosis model. A relative study was discussed for the case of a hypercube by Wang[19]. Furthermore, we have studied the strong local diagnosability property[7]. A given processor has the strong local diagnosability property if its local diagnosability equals its degree, where the degree is defined as the number of links inci-dent to this processor. A system has the strong local diagnosability property if every processor in it has this property. We prove that each processor in an n-dimensional star Snhas this strong local diagnosability property, and this property is

main-tained even if Snhas up to n  3 missing edges. The number n  3 is tight in the sense that the strong local diagnosability

property cannot be guaranteed if there are n  2 missing edges.

The remainder of this paper is organized as follows. In Section2, we present some definitions, notations, and terminol-ogies. The concept of local diagnosability for system diagnosis is also introduced in this section. Then, in Section3, we prove that an n-dimensional star keeps the strong local diagnosability property even if there exist n  3 missing edges in it. Finally, some conclusions are presented in Section4.

2. Preliminaries and local diagnosability

The topology of a multiprocessor system can be modeled as an undirected graph G = (V, E), where the set of nodes V rep-resents the set of all processors and the set of edges E reprep-resents the set of all connecting links between the processors. Let G0

be a subgraph of G and

v

be a node in G0_{; then, deg}

G0ð

v

Þ denotes the degree of

v

in subgraph G0. The neighborhood set of a node

v

, denoted by N(

v

), is defined as the set of all nodes adjacent to

v

.

Let n be a positive integer and hni be the set {1, 2 . . . n}. An n-dimensional star[1], denoted by Sn, is a graph whose set of

nodes consists of all permutations on hni. Each node is uniquely assigned a label x1x2. . .xn, where xi2 hni for 1 6 i 6 n and

xi–xjfor i – j. Each node x1x2. . .xi1xixi+1. . .xnis adjacent to the nodes xix2. . .xi1x1xi+1. . .xnfor 2 6 i 6 n, that is, nodes

ob-tained by the transposition of the first coordinate with the ith coordinate of the node. Consequently, there exist n! nodes in an n-dimensional star, and each node has degree n  1. Let x = x1x2. . .xnbe a node in an n-dimensional star Sn. We use (x)ito

denote the ith coordinate xiof x for 1 6 i 6 n. We say that two nodes x and y in Snare adjacent to each other with an ith edge

or an edge in dimension i if x can be obtained by the transposition of the first coordinate with the ith coordinate of y. Then, x is said to be the ith neighbor of y and it is denoted as x = yi_{, and vice versa. In addition, we use S}i

nto denote the subgraph of Sn

that is induced by the nodes x0_{s with (x)}

n= i for 1 6 i 6 n. Thus, Sncan be decomposed into n subgraphs Sinfor 1 6 i 6 n and

each Si

nis isomorphic to Sn1.

Under the comparison model[13,14], a system performs diagnosis by the specific procedure described below. For each processor w linked to two distinct processors u and

v

, the diagnosis is performed by simultaneously sending two identical signals from w to u and from w to

v

, and then comparing their returning responses. The comparison result of w for the two responses from u and

v

is denoted by r((u,

v

)w). An agreement is denoted by r((u,

v

)w) = 0, whereas a disagreement is denoted

by r((u,

v

)w) = 1. Because the comparator processor might be faulty, if r((u,

v

)w) = 1, at least one member of {u,

v

, w} is faulty;

or, if r((u,

v

)w) = 0 and w is known to be fault-free, both u and

v

are fault-free. Furthermore, a special case of the comparison

model, called the MM⁄_{model[17], assumes that a comparison is performed by each processor for each pair of distinct}

con-nected neighbors.

A labeled multigraph M = (V, C), called a comparison graph, is usually used to model this diagnosis strategy, where V rep-resents the set of all processors in G and C reprep-resents the set of labeled edges. Each labeled edge (u,

v

)w2 C implies that

pro-cessors u and

v

are being compared by processor w.

The collection of all test results of a test assignment is called a syndrome. Formally, a syndrome is a function

r

:C ? {0, 1}. For a given syndrome

r

, a subset of processors F  V(G) is said to be consistent with

r

if the syndrome

r

can be produced when all processors in F are faulty and all processors in V  F are fault-free. Let

r

Fdenote the set of syndromes that are

con-sistent with F. We say that a system is diagnosable if for every syndrome

r

, a unique set of processors F  V is consistent with it. A system is defined to be t-diagnosable if the system is diagnosable as long as the number of faulty processors does not exceed t. In other words, a system is t-diagnosable if given a test syndrome

r

Fproduced by the system under the presence of

a set of faulty nodes F with jFj 6 t, any set of faulty nodes F0 _{consistent with}

_r

Fwith jF0j 6 t must be F0= F. The maximum

number t for which a system is t-diagnosable is called the diagnosability of the system. Two distinct subsets of processors F1, F2 V are distinguishable if and only if every syndrome consistent with F1differs from that consistent with F2.

The next lemma follows trivially from the definition of the t-diagnosability of a system.

Lemma 1 [17]. A system G(V, E) is t-diagnosable if and only if, for each pair of distinct set of nodes (F1, F2) with jF1j, jF2j 6 t, (F1, F2)

is a distinguishable pair.

The following lemma is a useful characterization for the distinguishability of two sets of nodes under the comparison model.

Lemma 2 [17]. Let F1and F2be two distinct subsets of nodes. (F1, F2) is a distinguishable pair if and only if at least one of the

following conditions is satisfied: (SeeFig. 1for an illustration.)

(1) $u, w 2 V  F1 F2and $v 2 (F1 F2) [ (F2 F1) such that (u, v)w2 C.

(2) $u,

v

2 F1 F2and $w 2 V  F1 F2such that (u, v)w2 C, or

(3) $u,

v

2 F2 F1and $w 2 V  F1 F2such that (u, v)w2 C.

In contrast to the global sense in system diagnosis, Chiang and Tan[3]present a local concept called the local diagnos-ability of a given node in a system. This method requires only the correct identification of the faulty or fault-free status of a single node. Below are two definitions that introduce the concept of local diagnosability.

Definition 1 (3). A system G(V, E) is locally t-diagnosable at node x if, given a test syndrome

r

Fproduced by the system

under the presence of a set of faulty nodes F containing node x with jFj 6 t, every set of faulty nodes F0_{consistent with}

_r

Fand

jF0_{j 6 t, must also contain node x.}

Definition 2 (3). The local diagnosability tl(x) of a node x in a system G(V, E) is defined to be the maximum number of t for G

being locally t-diagnosable at x, that is, tl(x) = max{tjG is locally t  diagnosable at x}.

The close relationship between the local diagnosability and the traditional diagnosability is stated as follows. Lemma 3 (3). A system G(V, E) is t-diagnosable if and only if G is locally t-diagnosable at every node.

Lemma 4 (3). The diagnosability t(G) of a system G(V, E) is equal to the minimum value among the local diagnosability of every node in G, that is, t(G) = min{tl(x)jfor all x 2 V(G)}.

Under the comparison diagnosis model, an extended star structure for guaranteeing the local diagnosability of a given node is stated as below.

Definition 3 (3). Let x be a node in a graph G(V, E). For n 6 degG(x), we define an extended star ES(x; n) of order n at node x

with the node set V(ES(x; n)) = {x} [ {

v

ijj1 6 i 6 n,1 6 j 6 4} and the edge set E(ES(x; n)) = {(x,

v

k1), (

v

k1,

v

k2), (

v

k2,

v

k3),

(

v

k3,

v

k4)j1 6 k 6 n}. (SeeFig. 2for an illustration.)

We say that there exists an extended star structure ES(x; n) # G at node x if G contains an extended star ES(x; n) of order n at node x as a subgraph. Node x is called the root of ES(x; n). The extended star is a useful structure for computing the local diagnosability of a given node.

Lemma 5 (3). Let x be a node in a system G(V, E) with degG(x) = n. The local diagnosability of x is n if there exists an extended star

ES(x; n) # G at x.

Consequently, for every processor in a regular recursively constructed system, its local diagnosability can be easily deter-mined by finding an extended star structure at this processor. Moreover, the diagnosability of the entire system can also be obtained accordingly.

3. Strong local diagnosability property

In this section, we discuss the strong local diagnosability property[7]; this property describes the equivalence of the local diagnosability of a node and its degree. We prove that an n-dimensional star has this property, and it keeps this strong prop-erty even if there exists a bounded amount of missing edges.

Definition 4 (7). Let x be a node in a graph G(V, E). Node x has the strong local diagnosability property if the local diagnosability of x equals to its degree in G. That is, tl(x) = degG(x).

Definition 5 (7). Let G(V, E) be a graph. Graph G has the strong local diagnosability property if the local diagnosability of every node equals to its degree in G. That is, tl(x) = degG(x), for all x 2 V(G).

In the following, we show that an n-dimensional star with n P 4 has the strong local diagnosability property.

Lemma 6. For each node x in an n-dimensional star Snwith n P 4, there exists an extended star ES(x; n  1) # Snof order n  1

at x.

Proof. We use the notations mentioned inDefinition 3to find an extended star ES(x; n  1) as a subgraph of an n-dimen-sional star Sn at a given node x. Because Snis node symmetric, we arbitrarily choose x = x1x2. . .xnto be the root of an

ES(x;n  1).

For n = 4, we can find an extended star ES(x; 3) of order 3 at node x = x1x2x3x4(as shown inFig. 3), where the set of nodes

contains x,

v

11= x2x1x3x4,

v

12= x3x1x2x4,

v

13= x4x1x2x3,

v

14= x2x1x4x3,

v

21= x3x2x1x4,

v

22= x4x2x1x3,

v

23= x2x4x1x3,

v

24=

x3x4x1x2,

v

31= x4x2x3x1,

v

32= x2x4x3x1,

v

33= x3x4x2x1, and

v

34= x4x3x2x1, and the set of edges is {(x,

v

k1), (

v

k1,

v

k2),

(

v

k2,

v

k3), (

v

k3,

v

k4)j1 6 k 6 3}.

Suppose the result holds for all Sn1, for some n P 5. Now we claim the result also holds for Sn, that is, there is an extended

star ES(x; n  1) # Sn of order n  1 at each node x 2 V(Sn). Since the definition of star graphs, an Sn can be seen as a

composition of n Sn1’s. Let Sn1(x) be the subgraph of Sninduced by all nodes z0s where their nth coordinates are the same as

that of x, that is, (z)n= (x)n. By the assumption, there exists an ES(x; n  2) # Sn1(x) at node x. Pick the nth neighbor of x

denoted xn_{. Let S}

n1(xn) be the subgraph of Snthat xnbelongs to, in which all nodes has the same nth coordinate as xn. We can

v₁₁ v₁₂ v₁₃ v₁₄ v₂₁ v₂₂ v₂₃ v₂₄ v_n1 v_n2 v_n3 v_n4 x

Fig. 2. Extended star structure ES(x; n) of order n.

x=x

₁

x

₂

x

₃

x

₄

v

₁₁

=x

₂

x

₁

x

₃

x

₄

v

₁₂

=x

₃

x

₁

x

₂

x

₄

v

₁₃

=x

₄

x

₁

x

₂

x

₃

v

₁₄

=x

₂

x

₁

x

₄

x

₃

v

₂₁

=x

₃

x

₂

x

₁

x

₄

v

₂₂

=x

₄

x

₂

x

₁

x

₃

v

₂₃

=x

₂

x

₄

x

₁

x

₃

v

₂₄

=x

₃

x

₄

x

₁

x

₂

v

₃₁

=x

₄

x

₂

x

₃

x

₁

v

₃₂

=x

₂

x

₄

x

₃

x

₁

v

₃₃

=x

₃

x

₄

x

₂

x

₁

v

₃₄

=x

₄

x

₃

x

₂

x

₁

easily find another three different nodes a, b, and c, all in Sn1(xn), such that xn, a, b, and c form a path in Sn1(xn). The reason

is that the girth of Sn1is six and each node in it has degree n  2, which is larger than two when n P 5. Consequently, there

exists an extended star ES(x;n  1) # Snof order n  1 at each node x 2 V(Sn) for n P 4. h

Theorem 1. Let Snbe an n-dimensional star and n P 4. Each node x in Snhas the strong local diagnosability property and graph Sn

has the strong local diagnosability property.

Proof. ByLemmas 5 and 6, the local diagnosability of each node x 2 V(Sn) is n  1, because the degree of x in Snis n  1 and

there exists an extended star ES(x; n  1) of order n  1 at x for n P 4. Thus, every node in an n-dimensional star Snwith

n P 4 has the strong local diagnosability property. Therefore, graph Snhas the strong local diagnosability property. h

By the theorem above, we conclude that the diagnosability of Snis n  1 for n P 4; this is the same result as that obtained

by Zheng et al.[20].

In some circumstances, some links in a multiprocessor system may be missing. A missing edge indicates a link between two processors that has broken or failed for some reason. The existence of missing edges in a system may reduce the diag-nosability of the entire system and change the local diagdiag-nosability of each node in some manner. Meanwhile, the presence of missing edges changes the degrees of some of the nodes within the system. More precisely, in a regular graph, nodes adja-cent to some missing edges have lower degrees than other nodes. Accordingly, with a small number of missing edges, the nodes connecting to these edges may not keep the strong local diagnosability property, and the graph may not keep the strong local diagnosability property as well. Therefore, these new degrees will be used to decide whether the incomplete graph keeps the strong local diagnosability property.

In the following, we show that an n-dimensional star Snkeeps the strong local diagnosability property even with up to

n  3 missing edges for n P 4.

Before proving this claim, we present an example to show that an n-dimensional star Snmay not keep the strong local

diagnosability property if there exist n  2 missing edges. For an arbitrary node x in Sn, x is labeled as a permutation on

hni. Suppose there exist n  2 missing edges in Snthat are incident to node x (as shown inFig. 4). Then, the remaining degree

of x in this incomplete star with missing edges is 1. Let y be the only node adjacent to x. Let F1 be the set of nodes

{y} [ N(y)  {x} with jF1j = n  1, and F2be the set of nodes N(y) with jF2j = n  1. ByLemma 2, (F1, F2) is not a distinguishable

pair under the comparison diagnosis model, and this incomplete star with missing edges is not (n  1)-local diagnosable at y. Because the local diagnosability of y (which is less than n  1) does not equal its degree (which is n  1) in this incomplete star graph Sn, node y has no strong local diagnosability property anymore. Therefore, an incomplete star Snwith n  2

miss-ing edges cannot be guaranteed to have the strong local diagnosability property.

We now prove that an n-dimensional star Snstill keeps the strong local diagnosability property, provided that the number

of missing edges is at most n  3 for n P 4. Note that for a given set of edges L # E(G) in a system G, we use G  L to denote the subgraph with node set V(G) and edge set E(G)  L.

Lemma 7. Let Snbe an n-dimensional star with n P 4, and let F be an arbitrary set of missing edges with jFj 6 n  3. For each node

x in Sn, there exists an extended star ESðx; degSnFðxÞÞ # Sn F at x, where degSnFðxÞ denotes the remaining degree of node x in Sn F.

Proof. We prove this lemma by induction on n.

For the base case n = 4, each node in S4is labeled as a permutation on h4i. Let x = x1x2x3x4be any node in S4. We first

construct two extended star structures of order 3 around node x; one of which is the same as that described in the proof ofLemma 6 (as shown in Fig. 5(a)), and the other one contains the node set {x,

v

11= x2x1x3x4,

v

12= x4x1x3x2,

v

13=

x1x4x3x2,

v

14= x3x4x1x2,

v

21= x3x2x1x4,

v

22= x2x3x1x4,

v

23= x1x3x2x4,

v

24= x4x3x2x1,

v

31= x4x2x3x1,

v

32= x3x2x4x1,

v

33= x1x2x4x3,

v

34

= x2x1x4x3} and the edge set {(x,

v

k1), (

v

k1,

v

k2), (

v

k2,

v

k3), (

v

k3,

v

k4)j1 6 k 6 3} (as shown inFig. 5(b)). It is easy to check that in

both structures, except for the 2nd, 3rd, or 4th edge of x, all edges are different.

x

x x

y

F1 F2

Since n  3 = 1, we only need to consider two situations in which the number of missing edges is 0 or 1. If there exist no missing edges in S4, an ES(x; 3) at x indeed exists. If there exists one missing edge in S4, one of two following cases may occur:

(1) if the missing edge is the 2nd, 3rd, or 4th edge of x, the degree of x is 2 and there exists an ES(x; 2) at x; (2) otherwise, we can pick any one of the two above structures to avoid the missing edge, in order to form an extended star ES(x; 3) at x. As a consequence, there exists an extended star ESðx; degSnFðxÞÞ # Sn F at each node x for n = 4 and jFj = 0 or 1.

For induction hypothesis, we suppose that the result is true for Sn1, for some n P 5. That is, for any set of missing edges F

with jFj 6 (n  3)  1, there exists an ESðx; degSn1FðxÞÞ # Sn1 F at each node x 2 V(Sn1)  F.

Now, we claim that the result also holds for Sn, for all jFj 6 n  3. We shall prove that for a set of missing edges F with

jFj 6 n  3, there exists an ESðx; degSnFðxÞÞ # Sn F at each node x 2 V(Sn)  F. Assume that the number of missing edges is at most n  3 in an n-dimensional star Snfor n P 5. Let f = (u,

v

) be an arbitrary missing edge. Because the star graph is edge

symmetric, without lost of generality, we let

v

= un. The n-dimensional star Sncan be seen as the composition of n subgraphs

Sk

nfor 1 6 k 6 n, where Sknis a subgraph of Sninduced by the nodes z0s with (z)n= k. Thus, the number of all missing edges F

except f in Snis at most n  4. Consider a node x in Sn; x is in one of the n induced subgraphs Skn; 1 6 k 6 n, and each S k nis

isomorphic to an (n  1)-dimensional star Sn1. Let Sn1(x) be the substar that x belongs to. By the induction hypothesis,

there exists an extended star ESðx; degSn1ðxÞF0ðxÞÞ # Sn1ðxÞ  F

0_{at x, where F}0_{is the set of all missing edges in S}

n1(x) and

jF0_{j 6 n  4.}

If the nth edge of x is missing (Fig. 6(a)), the degree of x in Sn1(x)  F0is equal to the degree of x in this incomplete star

Sn F with at most n  3 missing edges. If the nth edge of x is not missing (Fig. 6(b)), x is adjacent to its nth neighbor,

denoted by xn_{, through the nth edge. Let S}

n1(xn) be the subgraph that xnbelongs to. Since jF  fj 6 n  4, each node in

Sn1(xn)  F is adjacent to at least two other nodes in Sn1(xn)  F. We note again that in a star graph Sn1, each node has

degree n  2. Then, xn_{is adjacent to a node a in S}

n1(xn)  F, a is adjacent to another node b in Sn1(xn)  F, and b is adjacent

to another node c other than xn_{, a in S}

n1(xn)  F since the girth of Sn1is six and the degree of each vertex in Sn1(xn)  F is

not less than two. As a result, for n P 5, there exists an ESðx; degSnFðxÞÞ # Sn F at x for all jFj 6 n  3. The proof is

complete. h

Theorem 2. Let Snbe an n-dimensional star and n P 4, and let F be an arbitrary set of missing edges with jFj 6 n  3. For each

node x in Snwith missing edges F, node x has the strong local diagnosability property and graph Sn F has the strong local

diag-nosability property. x=x1x2x3x4 v11=x2x1x3x4 v₁₂=x₃x₁x₂x₄ v₁₃=x₄x₁x₂x₃ v14=x2x1x4x3 v₂₁=x₃x₂x₁x₄ v22=x4x2x1x3 v₂₃=x₂x₄x₁x₃ v₂₄=x₃x₄x₁x₂ v31=x4x2x3x1 v₃₂=x₂x₄x₃x₁ v₃₃=x₃x₄x₂x₁ v34=x4x3x2x1

(a)

(b)

x=x₁x₂x₃x₄ v₁₁=x₂x₁x₃x₄ v₁₂=x₄x₁x₃x₂ v13=x1x4x3x2 v₁₄=x₃x₄x₁x₂ v₂₁=x₃x₂x₁x₄ v22=x2x3x1x4 v₂₃=x₁x₃x₂x₄ v₂₄=x₄x₃x₂x₁ v₃₁=x₄x₂x₃x₁ v₃₂=x₃x₂x₄x₁ v33=x1x2x4x3 v₃₄=x₂x₁x₄x₃

Fig. 5. Two possible extended stars ES(x; 3) at any node x = x1x2x3x4described in the proof ofLemma 7.

x

x

n a b c

u

v

x

n

f

x

u

v

f

(a)

(b)

X

X

X

S

_n-1

(x

n

₎

S

_n-1

(x )

S

_n-1

(x

n

₎

S

_n-1

(x )

Proof. ByLemmas 5 and 7, the local diagnosability of each node x in an incomplete n-dimensional star Sn F is equal to its

remaining degree for n P 4 and jFj 6 n  3. Thus, every node in Sn F has the strong local diagnosability property.

Conse-quently, graph Sn F has the strong local diagnosability property. h

4. Conclusions

In this paper, we studied the system diagnosis of an n-dimensional star under the comparison model. Following the con-cept of local diagnosability and the extended star structure proposed by Chiang and Tan[3], the diagnosability of a system can be determined in a straightforward manner. By the definition of the strong local diagnosability property[7], we proved that an n-dimensional star has this property, and it keeps this strong property even if there exist up to n  3 missing edges in it. As a result, the diagnosability of an incomplete n-dimensional star system with arbitrary missing links can be obtained as the minimum value among the remaining degree of every processor, provided that the cardinality of the set of missing links does not exceed n  3.

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