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The g-good neighbor conditional diagnosability of twisted hypercubes under the PMC and MM* model
Applied Mathematics and Computation 332 (2018) 484–492
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The g-good neighbor conditional diagnosability of twisted hypercubes under the PMC and MM∗ model Huiqing Liu a, Xiaolan Hu b,a,∗, Shan Gao a a
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistic, Hubei University, Wuhan 430062, PR China School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, PR China
b
a r t i c l e
i n f o
Keywords: Twisted hypercubes g-good neighbor Rg -vertex-connectivity Conditional diagnosability PMC model MM∗ model
a b s t r a c t Connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks. The g-good-neighbor conditional faulty set is a special faulty set that every fault-free vertex should have at least g fault-free neighbors. The Rg -vertex-connectivity of a connected graph G is the minimum cardinality of a g-good-neighbor conditional faulty set X⊆V(G) such that G − X is disconnected. The g-good-neighbor conditional diagnosability is a metric that can give the maximum cardinality of g-good-neighbor conditional faulty set that the system is guaranteed to identify. The twisted hypercube is a new variant of hypercubes with asymptotically optimal diameter introduced by X.D. Zhu. In this paper, we first determine the Rg -vertex-connectivity of twisted hypercubes, then establish the g-good neighbor conditional diagnosability of twisted hypercubes under the PMC model and MM∗ model, respectively. © 2018 Elsevier Inc. All rights reserved.
1. Introduction A multiprocessor system comprises two or more processors, and various processors exchange information via links between them. With the rapid development of science and technology, the number of processors is increasing. As the size of multiprocessor systems increase, processor failure is inevitable. Therefore, the topic of reliability and fault-tolerance of multiprocessor systems has become an active area of intensive research. To evaluate the reliability of multiprocessor systems, fault diagnosability has become an important metric. There are several different diagnosis models being proposed to determine the diagnosability of a system. In this paper, we use the PMC model introduced by Preparata et al. [18] and MM∗ model suggested by Sengupta and Dahbura [19]. In the PMC model, every processor can test the processor that is adjacent to it. In the MM∗ model, every processor must test two processors if the processor is adjacent to the latter two processors. Many researchers have applied the PMC model and MM∗ model to identify faults in various topologies, see [1–3,9,16]. The diagnosability for multiprocessor systems assumes that all the neighbors of any processor may fail simultaneously. Therefore, for a system, its diagnosability is restricted by its minimum degree. However, the probability that this event occurs is very small in large-scale multiprocessor systems. To obtain a more practical diagnosability, Lai et al. [11] introduced
∗ Corresponding author at: School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, PR China. E-mail address: [email protected] (X. Hu).
https://doi.org/10.1016/j.amc.2018.03.042 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.
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conditional diagnosability under the assumption that all the neighbors of any processor in a multiprocessor system cannot be faulty at the same time. The conditional diagnosability of interconnection networks has been extensively investigated, see [2,4,7–10,25,30]. Recently, Peng et al. [17] proposed a new measure for fault diagnosis of the system, namely, g-good-neighbor conditional diagnosability, which requires that every fault-free node contains at least g fault-free neighbors. Compared with diagnosability, the g-good-neighbor conditional diagnosability improves accuracy in measuring the reliability of interconnection networks in heterogeneous environments. Meanwhile, Peng et al. [17] studied the g-good-neighbor conditional diagnosability of the n-dimensional hypercube Qn under the PMC model. Wang and Han [22] determined the g-good-neighbor diagnosability of the n-dimensional hypercube Qn under the MM∗ model. Since then, numerous studies have been investigated under the PMC model and MM∗ model, see [5,13,14,21,23,24,27,28]. In this paper, we consider the g-good-neighbor conditional diagnosability of the n-dimensional twisted hypercube Hn , which is a new variant of hypercubes with asymptotically optimal diameter introduced by Zhu [29], under the PMC model and MM∗ model, respectively. Our main results are listed below. Theorem 1.1. Let n be an integer with n ≥ 4. Then the g-good-neighbor conditional diagnosability of Hn under the PMC model is
tg (Hn ) =
2g ( n − g + 1 ) − 1 2n−1 − 1
if 0 ≤ g ≤ n − 3, if g − 2 ≤ g ≤ n − 1.
Theorem 1.2. Let n be an integer with n ≥ 5. Then the g-good-neighbor conditional diagnosability of Hn under the MM∗ model is
tg (Hn ) =
2g ( n − g + 1 ) − 1 2n−1 − 1
if 0 ≤ g ≤ n − 3, if g − 2 ≤ g ≤ n − 1.
The rest of this paper is organized as follows: Section 2 provides preliminaries for our notations, diagnosing a system and the twisted hypercubes. In Section 3, we determine the Rg -vertex-connectivity of twisted hypercubes. In Sections 4, we establish the g-good-neighbor conditional diagnosability of twisted hypercubes under the PMC model and MM∗ model, respectively. Our conclusions are given in Sections 5. 2. Preliminaries 2.1. Notations Let G = (V (G ), E (G )) be a simple and finite graph. The neighborhood NG (v ) of a vertex v is the set of vertices adjacent to v and the closed neighborhood of v is NG [v] = NG (v ) ∪ {v}. The degree dG (v ) of v is |NG (v )|. The minimum degree of G is denoted by δ (G). If dG (v ) = k for any v ∈ V (G ), then G is called a k-regular graph. For S⊆V(G), G[S] denotes the subgraph induced by S. The neighborhood set of S is defined as NG (S ) = (∪v∈S NG (v )) − S, and the closed neighborhood set of S is defined as NG [S] = NG (S ) ∪ S. We will use G − S to denote the subgraph G[V (G ) − S]. For any v ∈ V (G ), NS (v ) denotes the neighborhood of v in S. For two disjoint subsets S, T of V(G), let EG (S, T ) = {uv ∈ E (G ) | u ∈ S, v ∈ T }. The symmetric difference of F1 ⊆V(G) and F2 ⊆V(G) is defined as the set F1 F2 = (F1 − F2 ) ∪ (F2 − F1 ). Let Kn be a complete graph of order n. The minimum cardinality of a vertex set S⊆V(G) such that G − S is disconnected or has only one vertex, denoted by κ (G), is the connectivity of G. A subset F⊆V(G) is called a Rg -vertex-set of G if δ (G − F ) ≥ g. A Rg -vertex-cut of a connected graph G is a Rg -vertex-set F such that G − F is disconnected. The Rg -vertex-connectivity of G, denoted by κ g (G), is the cardinality of a minimum Rg -vertex-cut of G. Note that κ 0 (G ) = κ (G ). 2.2. The PMC model and MM∗ model for diagnosis A multiprocessor system is typically represented by an undirected simple graph G = (V, E ), where V(G) stands for the processors and E(G) represents the links between two processors. Preparata et al. [18] proposed the PMC model, which performs diagnosis by testing the neighboring processor via the links between them. Under the PMC model, tests can be performed between any two adjacent vertices u and v. We use the ordered pair (u, v ) to denote a test that u diagnoses v. The result of a test (u, v ) is reliable if and only if u is fault-free. In this condition, the result is 0 if v is fault-free, and is 1 otherwise. A test assignment for a system G is a collection of tests, which can be represented a directed graph T = (V (G ), L ), where V(G) is the vertex set of G and L = {(u, v ) | u, v ∈ V (G ) and uv ∈ E (G )}. The collection of all test results from the test scheme T is termed as a syndrome σ : L → {0, 1}. Let T = (V (G ), L ) be a test assignment, and F a subset of V(G). For any given syndrome σ resulting from T, F is said to be consistent with σ if the syndrome σ can be produced when all vertices in F are faulty and all vertices in V (G ) − F are fault-free. That is, if u is fault-free, then σ ((u, v )) = 0 if v is fault-free, and σ ((u, v )) = 1 if v is faulty; if u is faulty, then σ ((u, v )) can be either 0 or 1 no matter v is faulty or not. Therefore, a faulty set F may be consistent with different syndromes. We use σ (F) to represent the set of all possible syndromes with which the faulty set F can be consistent.
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Fig. 1. Illustration of a distinguishable pair under the PMC model.
Fig. 2. Illustration of a distinguishable pair under the MM∗ model.
Let F1 and F2 be two distinct faulty sets of V(G), F1 and F2 are distinguishable if σ (F1 ) ∩ σ (F2 ) = ∅; otherwise, F1 and F2 are indistinguishable. In other words, if σ (F1 ) ∩ σ (F2 ) = ∅, then (F1 , F2 ) is a distinguishable pair; otherwise, (F1 , F2 ) is an indistinguishable pair. A system G is called t-diagnosable, if any two distinct faulty sets F1 , F2 ⊆V(G) are distinguishable, provided that |F1 |, |F2 | ≤ t. The diagnosability of G, denoted by t(G), is the maximum t such that G is t-diagnosable. In [6], Dahbura and Masson proposed a sufficient and necessary condition of t-diagnosable systems. Theorem 2.1. [6] A system G = (V, E ) is t-diagnosable if and only if, for any two distinct subsets F1 and F2 of V(G) with |F1 |, |F2 | ≤ t, there is at least one test from V (G ) − F1 − F2 to F1 F2 . For the PMC model, Peng et al. [17] proposed the following theorem. Theorem 2.2. [17] For any two distinct subsets F1 and F2 of V(G), (F1 , F2 ) is a distinguishable pair under the PMC model if and only if there is a vertex u ∈ V (G ) − F1 − F2 and a vertex v ∈ F1 F2 such that uv ∈ E (G ) (see Fig. 1). Sengupta and Dahbura [19] proposed the MM∗ model, which performs diagnosis by sending the same inputs to a pair of adjacent processors, and comparing their responses. In the MM∗ model, tests can be performed from any vertex w to its any two neighbors u and v. We use the labeled pair (u, v )w to denote a test performed by w on its neighbors u and v. The result of a test (u, v )w is reliable if and only if w is fault-free. In this condition, the result is 0 if both u and v are fault-free, and is 1 otherwise. The test scheme of a system G is often represented by a multigraph M = (V (G ), L ), where L = {(u, v )w | u, v, w ∈ V (G ) and uw, vw ∈ E (G )}. Since a pair of vertices may be compared by different vertices, M is a multigraph. The collection of all test results from the test scheme M is termed as a syndrome σ : L → {0, 1}. Given a subset of vertices F⊆V(G), we say that F is consistent with σ if the syndrome σ can be produced when all nodes in F are faulty and all nodes in V (G ) − F are fault-free. That is, if w is fault-free, then σ ((u, v )w ) = 0 if u and v are fault-free, and σ ((u, v )w ) = 1 otherwise; if w is faulty, then σ ((u, v )w ) can be either 0 or 1 no matter u and v are faulty or not. Sengupta and Dahbura [19] proposed a sufficient and necessary condition for two distinct subsets F1 and F2 to be a distinguishable pair under the MM∗ model. Theorem 2.3. [19] For any two distinct subsets F1 and F2 of V(G), (F1 , F2 ) is a distinguishable pair under the MM∗ model if and only if one of the following conditions is satisfied (see Fig. 2): (1) There are two vertices u, w ∈ V (G ) − F1 − F2 and there is a vertex v ∈ F1 F2 such that uw, vw ∈ E (G ), (2) There are two vertices u, v ∈ F1 − F2 and there is a vertex w ∈ V (G ) − F1 − F2 such that uw, vw ∈ E (G ), (3) There are two vertices u, v ∈ F2 − F1 and there is a vertex w ∈ V (G ) − F1 − F2 such that uw, vw ∈ E (G ). In the following, we will give the definition of g-good-neighbor conditional faulty set and the g-good-neighbor conditional diagnosability of a system. Definition 2.4. Let G = (V, E ) be a system. A faulty set F⊆V(G) is called a g-good-neighbor conditional faulty set if δ (G − F ) ≥ g. Definition 2.5. A system G = (V, E ) is g-good-neighbor conditional t-diagnosable, if any two distinct g-good-neighbor conditional faulty sets F1 , F2 ⊆V(G) are distinguishable, provided that |F1 |, |F2 | ≤ t. The g-good-neighbor conditional diagnosability of G, denoted by tg (G), is the maximum t such that G is g-good-neighbor conditionally t-diagnosable. Note that t0 (g) = t (G ). By definition of the g-good-neighbor conditional diagnosability, we have the following proposition which is also observed in [17].
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Fig. 3. H3 and H4 .
Proposition 2.6. [17] For any system G = (V, E ), if g ≤ g , then tg (G ) ≤ tg (G ) under the PMC model and MM∗ model, respectively. 2.3. The twisted hypercubes In this subsection, we first introduce the definition of the n-dimensional twisted hypercube Hn , and then give some properties of Hn . Denote by Z2n the set of binary strings of length n. For x, y ∈ Z2n , xy denotes the sum of x and y in the group Z2n , i.e., (x y )i = xi + yi (mod 2 ) (for x ∈ Z2n , xi denotes the ith bit of x). If x is a binary string of length n1 and y is a binary string of length n2 , then xy is the concatenation of x and y, which is a binary string of length n1 + n2 . If Z is a set of binary strings, then let xZ = {xy : y ∈ Z }. For x ∈ Z2n , we use x[i, j] to denote the binary string xi xi+1 . . . x j for 1 ≤ i < j ≤ n. We first present an integer function κ (n). Definition 2.7. Let κ be the integer function defined as the following:
κ (n ) =
0 max{1, log2 n − 2 log2 log2 n}
if n = 1, otherwise.
Next, a permutation φ of binary strings is given as follows. Definition 2.8. Assume x ∈ Z2n and n ≥ 2. Then φ (x ) ∈ Z2n is the binary string such that φ (x )[1, κ (n )] = x[1, κ (n )] x[n − κ (n ) + 1, n], φ ( x )[ κ ( n ) + 1 , n ] = x [ κ ( n ) + 1 , n ] . Note that the restriction of φ to Z2n is indeed a permutation of Z2n , with φ 2 (x ) = x. Now we give a recursive definition of the n-dimensional twisted hypercube Hn . Definition 2.9. Set H1 := K2 , with vertices 0 and 1. For n ≥ 2, Hn is obtained from two copies of Hn−1 , 0Hn−1 and 1Hn−1 , by adding edges connecting 0x and 1φ (x) for all x ∈ V (Hn−1 ). The vertex set of Hn is Z2n . It follows from the definition that H1 = K2 , H2 = C4 , H3 and H4 are depicted in Fig. 3. It is seen that Hn is a kind of n-dimensional hypercube-like networks. Hypercube-like network was first proposed by Vaidya et al. [20] for a unified study on hypercube variants. By the definition of Hn , we have the following result which is also observed in [29]. q
n−q
Proposition 2.10. For 1 ≤ q < n and b ∈ Z2 , the subgraph induced by bZ2
is a Hn−q .
Let a ∈ {0, 1} be an integer and a the complement of a, i.e., a = 1 − a. Let v = v1 v2 · · · vn be any vertex of Hn . Then v q has only one neighbor φ (v ) in v1 Hn−1 , and n − 1 neighbors in v1 Hn−1 . By Proposition 2.10, for 1 ≤ q ≤ n − 1 and b ∈ Z2 , the n−q subgraph induced by bZ2 is a Hn−q . So for every neighbor w = w1 w2 · · · wn of v in v1 Hn−1 , there exists some q (1 ≤ q ≤ n − 1 ) such that wi = vi for 1 ≤ i ≤ q. The following results will be used in Sections 3 and 4. Proposition 2.11. For any u, v ∈ V (Hn ), u and v have at most two common neighbors. Proof. We prove this proposition by induction on n. In the basis step, for n = 2, H2 is a 4-cycle. It is seen that any two vertices of H2 have at most two common neighbors. In the induction step, assume that the statement is true for n = k − 1. That is, any two vertices of Hk−1 have at most two common neighbors. When n = k, let u and v be two vertices of Hk .
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If u, v ∈ V (0Hk−1 ) or u, v ∈ V (1Hk−1 ), say u, v ∈ V (0Hk−1 ), then u and v have at most two common neighbors in 0Hk−1 by induction hypothesis. Note that u and v have no common neighbors in 1Hk−1 . Then u and v have at most two common neighbors. Now we assume u ∈ V (0Hk−1 ) and v ∈ V (1Hk−1 ). If v = φ (u ), then u and v have no common neighbors. So we can assume v = φ (u ). Then the common neighbor(s) of u and v will be φ (u) and/or φ −1 (v ). Therefore u and v have at most two common neighbors. Proposition 2.12. [15] Let Q be a subgraph of Hn . If δ (Q) ≥ g, then |V(Q)| ≥ 2g . For any subgraph Q of Hn , we use NHn [Q] to denote the closed neighborhood set of V(Q) in Hn . Proposition 2.13. [26] Let Q be a subgraph of Hn . If δ (Q) ≥ g, then |NHn [Q]| ≥ 2g (n − g + 1 ). For convenience, we write the consecutive k 0’s in the binary string as 0k . Proposition 2.14. Let X = {v1 v2 · · · vg 0n−g | vi ∈ {0, 1}, 1 ≤ i ≤ g} ⊂ V (Hn ) and F1 = NHn (X ). Then Hn [X] is a g-regular graph, and |F1 | = 2g (n − g). Proof. Let v = v1 v2 · · · vg 0n−g be any vertex of X, then NF1 (v ) = {v1 v2 · · · vg 0n−g− j−1 10 j | 0 ≤ j ≤ n − g − 1}. Thus F1 = n−g− j−1 10 j | v ∈ {0, 1}, 1 ≤ i ≤ g}. It is seen that v has g neighbors in X and n − g neighbors in F . 1 i 0≤ j≤n−g−1 {v1 v2 · · · vg 0 Then Hn [X] is a g-regular graph. Note that |X | = 2g , and any two different vertices u, v ∈ X have no common neighbors in F1 . Then |F1 | = 2g (n − g). 3. The Rg -vertex-connectivity of Hn In order to obtain the g-good-neighbor conditional diagnosability of Hn , we have to investigate the Rg -vertex-connectivity of Hn , which is closely related to g-good-neighbor conditional diagnosability proposed by Latifi et al. [12]. As a more refined index than the traditional connectivity, the Rg -vertex-connectivity can be used to measure the conditional fault tolerance of networks. In this section, we determine the Rg -vertex-connectivity of Hn for 0 ≤ g ≤ n − 2. In [26], Ye and Liang considered the Rg -vertex-connectivity of any n-dimensional hypercube-like network and got the following lower bound. Lemma 3.1. [26] For 0 ≤ g ≤ n − 2, κ g (Hn ) ≥ 2g (n − g). Lemma 3.2. For 0 ≤ g ≤ n − 2, let X = {v1 v2 · · · vg 0n−g | vi ∈ {0, 1}, 1 ≤ i ≤ g} ⊂ V(Hn ), F1 = NHn (X ) and F2 = NHn [X]. Then F1 is a g-good-neighbor conditional faulty set and F2 is a max{g, n − 3}-good-neighbor conditional faulty set of Hn . Proof. By Proposition 2.14, Hn [X] is a g-regular graph. So we only need to show that each vertex in V (Hn ) − F2 has at least max{g, n − 3} neighbors out of F2 , that is, each vertex in V (Hn ) − F2 has at most min{n − g, 3} neighbors in F1 . Let u = u1 u2 · · · un−1 un be a vertex in V (Hn ) − F2 . Note that F1 = 0≤ j≤n−g−1 {v1 v2 · · · vg 0n−g− j−1 10 j | vi ∈ {0, 1}, 1 ≤ i ≤ g} ⊂ F2 , then u has t ≥ 2 bits equal 1 between the last n − g bits. That is, there exists t ≥ 2 such that uki = 1 and g + 1 ≤ ki ≤ n for any 1 ≤ i ≤ t. Assume that g + 1 ≤ k1 < · · · < kt ≤ n. If g = n − 2, then u = u1 u2 · · · un−2 11, thus u has 2 = n − g neighbors {u1 u2 · · · un−2 10, u1 u2 · · · un−2 01} in F1 . So we assume g ≤ n − 3. Let v = u1 u2 · · · ug 0n−g be a vertex in X. If all the neighbors of u in F1 have the same first g bits as that of u, then NF1 (u ) ⊆ {u1 u2 · · · ug 0n−g− j−1 10 j | 0 ≤ j ≤ n − g − 1} = NF1 (v ). By Proposition 2.11, u and v have at most two common neighbors, then u has at most 2 < n − g neighbors in F1 . So we assume u has a neighbor w = w1 w2 · · · wn ∈ F1 such that w has the same first q (0 ≤ q ≤ g − 1) bits as that of u. Then w = u1 u2 · · · uq uq+1 (uq+2 · · · uq+κ +1 un−κ +1 · · · un )uq+κ +2 · · · un , where κ = κ (n − q − 1 ). Note that w ∈ F1 , then for any 1 ≤ i ≤ t − 1, wki = uki un−q−κ +ki −1 = 0, thus un−q−κ +ki −1 = 1. Since n − q − 1 ≥ n − g ≥ 3 and n − q − 1 ≥ 2 log2 (n − q − 1 ) − log2 log2 (n − q − 1 ), then n − q − 1 ≥ 2κ , i.e., n − κ + 1 ≥ q + κ + 2. Note that ki ≥ g + 1 ≥ q + 2, then n − q − κ + ki − 1 ≥ n − κ + 1, thus un−q−κ +ki −1 = wn−q−κ +ki −1 for 1 ≤ i ≤ t − 1, hence t = 2. If wk2 = uk2 un−q−κ +k2 −1 = 0, then un−q−κ +k2 −1 = 1 and un−q−κ +k2 −1 = wn−q−κ +k2 −1 , a contradiction with w ∈ F1 . Therefore wk2 = 1 and n − q − κ + k1 − 1 = k2 . Hence u has only one neighbor of w such that w has the same first q (q is an integer satisfying (n − q − 1 ) − κ (n − q − 1 ) = k2 − k1 ) bits as that of u, all the other neighbors of u are in NF1 (v ). By Proposition 2.11, u has at most two neighbors in NF1 (v ). Thus u has at most 3 ≤ n − g neighbors in F1 . Therefore we complete the proof of Lemma 3.2. Theorem 3.3. For 0 ≤ g ≤ n − 2, κ g (Hn ) = 2g (n − g). Proof. Let X = {v1 v2 · · · vg 0n−g | vi ∈ {0, 1}, 1 ≤ i ≤ g} ⊂ V(Hn ) and F1 = NHn (X ). Clearly, Hn − F1 is disconnected. By Lemma 3.2, F1 is a g-good-neighbor conditional faulty set. Then F1 is a Rg -vertex-cut of Hn . By Proposition 2.14, |F1 | = 2g (n − g). Then κ g (Hn ) ≤ 2g (n − g). By Lemma 3.1, κ g (Hn ) ≥ 2g (n − g). Hence κ g (Hn ) = 2g (n − g). 4. The g-good-neighbor conditional diagnosability of Hn In this section, we will give some lemmas, then determine tg (Hn ) for 0 ≤ g ≤ n − 1 under the PMC model and MM∗ model, respectively.
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Lemma 4.1. For 0 ≤ g ≤ n − 2, let X = {v1 v2 · · · vg 0n−g | vi ∈ {0, 1}, 1 ≤ i ≤ g} ⊂ V(Hn ), F1 = NHn (X ) and F2 = NHn [X]. Then (F1 , F2 ) is an indistinguishable pair under the PMC model and MM∗ model, respectively. Proof. Note that F1 F2 = X and NHn (X ) = F1 . Then (F1 , F2 ) is an indistinguishable pair under the PMC model by Theorem 2.2, and under the MM∗ model by Theorem 2.3. From Propositions 2.12, 2.13, Lemmas 3.2, 4.1, and combining with the definition of tg , we have the following result. Theorem 4.2. For 0 ≤ g ≤ n − 2, tg (Hn ) ≤ 2g (n − g + 1 ) − 1 under the PMC model and MM∗ model, respectively. For 0 ≤ g ≤ n − 1, tg (Hn ) has the following upper bound. Theorem 4.3. For 0 ≤ g ≤ n − 1, tg (Hn ) ≤ 2n−1 − 1 under the PMC model and MM∗ model, respectively. Proof. Let F1 = V (0Hn−1 ) and F2 = V (1Hn−1 ). Then |F1 | = |F2 | = 2n−1 . Note that F1 and F2 are g-good-neighbor conditional faulty sets. Since F1 ∪ F2 = V (Hn ), then (F1 , F2 ) is an indistinguishable pair under the PMC model by Theorem 2.2, and under the MM∗ model by Theorem 2.3. Thus tg (Hn ) ≤ 2n−1 − 1. Lemma 4.4. Let F1 and F2 be two distinct g-good-neighbor conditional faulty sets in Hn such that |F1 |, |F2 | ≤ 2g (n − g + 1 ) − 1. For 0 ≤ g ≤ n − 3, we have F1 ∪ F2 = V(Hn ). Proof. Since n − g ≥ 3, we have 2n−g−1 ≥ n − g + 1. Then
|F1 ∪ F2 | ≤ |F1 | + |F2 | ≤ 2g+1 (n − g + 1 ) − 2 ≤ 2g+1 2n−g−1 − 2 < 2n . Thus F1 ∪ F2 = V(Hn ).
4.1. tg (Hn ) under the PMC model In this subsection, we will determine tg (Hn ) for n ≥ 4 and 0 ≤ g ≤ n − 1 under the PMC model. Lemma 4.5. Let 0 ≤ g ≤ n − 3. For any two distinct g-good-neighbor conditional faulty sets F1 and F2 in Hn with |F1 |, |F2 | ≤ 2g (n − g + 1 ) − 1, (F1 , F2 ) is a distinguishable pair under the PMC model. Proof. Suppose that (F1 , F2 ) is an indistinguishable pair. Assume, without loss of generality, that F2 − F1 = ∅. By Lemma 4.4, we have F1 ∪ F2 = V(Hn ). Since F1 = F2 , then F1 F2 = ∅. By Theorem 2.2, F1 ∩ F2 is a vertex cut of Hn . Since F1 and F2 are g-good-neighbor conditional faulty sets of Hn , F1 ∩ F2 is also a g-good-neighbor conditional faulty set which implies F1 ∩ F2 is a Rg -vertex-cut of Hn . By Theorem 3.3, |F1 ∩ F2 | ≥ 2g (n − g). Since F1 is a g-good-neighbor conditional faulty set, all the vertices in F2 − F1 have at least g neighbors out of F1 . By Theorem 2.2, EHn (F1 F2 , V (Hn ) − F1 − F2 ) = ∅, then δ (Hn [F2 − F1 ] ) ≥ g. Thus |F2 − F1 | ≥ 2g by Proposition 2.12. So
2g (n − g + 1 ) − 1 ≥ |F2 | = |F1 ∩ F2 | + |F2 − F1 | ≥ 2g (n − g) + 2g = 2g (n − g + 1 ), a contradiction.
By Theorem 4.2 and Lemma 4.5, we get the g-good neighbor of conditional diagnosability of Hn under the PMC model for n ≥ 4 and 0 ≤ g ≤ n − 3. Theorem 4.6. For n ≥ 4 and 0 ≤ g ≤ n − 3, tg (Hn ) = 2g (n − g + 1 ) − 1 under the PMC model. For n − 2 ≤ g ≤ n − 1, we have the following result. Theorem 4.7. For n ≥ 4 and n − 2 ≤ g ≤ n − 1, tg (Hn ) = 2n−1 − 1 under the PMC model. Proof. By Proposition 2.6, tg (Hn ) ≥ tn−3 (Hn ). Note that tn−3 (Hn ) = 2n−1 − 1 by Theorem 4.6, then tg (Hn ) ≥ 2n−1 − 1. Since tg (Hn ) ≤ 2n−1 − 1 by Theorem 4.3, then tg (Hn ) = 2n−1 − 1. 4.2. tg (Hn ) under the MM∗ model In this subsection, we will determine tg (Hn ) for n ≥ 5 and 0 ≤ g ≤ n − 1 under the MM∗ model. First we consider the case of g = 0, the 0-good-neighbor condition does not have any restriction on the faulty sets in this case, then t0 (Hn ) = t (Hn ). Theorem 4.8. For n ≥ 5, t0 (Hn ) = n under the MM∗ model. Proof. First we show that t0 (Hn ) ≤ n. Let v ∈ V (Hn ), F1 = NHn (v ) and F2 = NHn [v]. Note that F1 F2 = {v}, then (F1 , F2 ) is an indistinguishable pair by Theorem 2.3, thus t0 (Hn ) ≤ n by the definition of t0 . Now we show that t0 (Hn ) ≥ n. We need to show that for any two distinct faulty sets F1 , F2 in Hn with |F1 |, |F2 | ≤ n, (F1 , F2 ) is a distinguishable pair. Suppose to the contrary that (F1 , F2 ) is an indistinguishable pair. Assume, without loss of generality,
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that F2 − F1 = ∅, then |F2 − F1 | > 0. Note that |F1 ∪ F2 | ≤ |F1 | + |F2 | ≤ 2n < 2n for n ≥ 5, then F1 ∪ F2 = V(Hn ). Since F1 = F2 , then F1 F2 = ∅. By Theorem 2.1, F1 ∩ F2 is a vertex cut of Hn . By Theorem 3.3, |F1 ∩ F2 | ≥ n. So
n ≥ |F2 | = |F1 ∩ F2 | + |F2 − F1 | > |F1 ∩ F2 | ≥ n, a contradiction. Therefore, we complete the proof of Theorem 4.8.
In the following, we assume that 1 ≤ g ≤ n − 1. We will determine tg (Hn ) for 1 ≤ g ≤ n − 3 and g − 2 ≤ g ≤ n − 1, respectively. Lemma 4.9. For n ≥ 5 and 1 ≤ g ≤ n − 2. Let F1 and F2 be two distinct g-good-neighbor conditional faulty sets such that |F1 |, |F2 | ≤ 2g (n − g + 1 ) − 1. If F1 and F2 are indistinguishable under the MM∗ model, then Hn − F1 − F2 has no isolated vertex. Proof. Let W be the set of isolated vertices in Hn − F1 − F2 , and Y the subgraph induced by the vertex set V (Hn ) − F1 − F2 − W . We will show that W = ∅. Suppose to the contrary that W = ∅ and let w ∈ W . We consider the following two cases. Case 1. g = 1. In this case, |F1 |, |F2 | ≤ 2n − 1. Since F1 is a 1-good-neighbor conditional faulty set, w must have at least one neighbor out of F1 . Note that w is an isolated vertex of Hn − F1 − F2 , then NHn −F1 (w ) ⊆ F2 − F1 . Since F1 and F2 are indistinguishable, then |NHn −F1 (w )| = 1 by Theorem 2.3(3). Similarly, NHn −F2 (w ) ⊆ F1 − F2 and |NHn −F2 (w )| = 1. Therefore |NF1 ∩F2 (w )| = n − 2. So we have
|NF1 ∩F2 (w )| = (n − 2 )|W | ≤
w∈W
dHn (v ) = n|F1 ∩ F2 | ≤ n(|F1 | − 1 ) ≤ n(2n − 2 ).
v∈F1 ∩F2
Thus |W | ≤ n(2n − 2 )/(n − 2 ) ≤ 2n + 4 as n ≥ 5. If V (Y ) = ∅, then |V (Hn )| = |F1 ∪ F2 | + |W |. Note that n ≥ 5, then
2(2n − 1 ) ≥ |F1 | + |F2 | = |F1 ∪ F2 | + |F1 ∩ F2 | = |V (Hn )| − |W | + |F1 ∩ F2 | ≥ 2n − (2n + 4 ) + (n − 2 ) > 4n, a contradiction. So V(Y) = ∅. Since F1 and F2 are indistinguishable, F1 ∩ F2 is a R1 -vertex-cut of Hn by Theorem 2.3. By Theorem 3.3, |F1 ∩ F2 | ≥ 2n − 2. Note that for any w ∈ W, w has one neighbor in F1 − F2 and one neighbor in F2 − F1 . So we have |F1 ∪ F2 | = 2n − 2 which implies |F1 | = |F2 | = 2n − 1 and |F1 − F2 | = |F2 − F1 | = 1. Assume F1 − F2 = {v1 } and F2 − F1 = {v2 }. For any w ∈ W, we have v1 , v2 ∈ NHn (w ). By Proposition 2.11, |W| ≤ 2. By definition of Hn , Hn has no triangles, then w and vi have no common neighbors for i = 1, 2. If |W | = 1, then W = {w} and |NF1 ∪F2 (w )| = n − 2, |NF1 ∪F2 (v1 )| = n − 1 and |NF1 ∪F2 (v2 )| = n − 1. By Proposition 2.11, v1 and v2 have at most one common neighbor in F1 ∪ F2 . So
2n − 2 = |F1 ∪ F2 | ≥ (n − 2 ) + (n − 1 ) + (n − 2 ) = 3n − 5, i.e., n ≤ 3, a contradiction with n ≥ 5. If |W | = 2, denote W = {w1 , w2 }. Then |NF1 ∪F2 (wi )| = n − 2 for 1 ≤ i ≤ 2, |NF1 ∪F2 (v1 )| = n − 2 and |NF1 ∪F2 (v2 )| = n − 2. By Proposition 2.11, v1 and v2 have no common neighbors in F1 ∪ F2 , w1 and w2 have no common neighbors in F1 ∪ F2 . So
2n − 2 = |F1 ∪ F2 | ≥ (n − 2 ) + (n − 2 ) + (n − 2 ) + (n − 2 ) = 4n − 8, i.e., n ≤ 3, a contradiction with n ≥ 5. Case 2. g ≥ 2. Since F1 is a g-good-neighbor conditional faulty set, then |NHn −F1 (w )| ≥ g. Note that w is an isolated vertex in Hn − F1 − F2 , then NHn −F1 (w ) ⊆ F2 − F1 . Since g ≥ 2, by Theorem 2.3(3), F1 and F2 are distinguishable, a contradiction. Lemma 4.10. Let n ≥ 5 and 1 ≤ g ≤ n − 3. For any two distinct g-good-neighbor conditional faulty sets F1 and F2 with |F1 |, |F2 | ≤ 2g (n − g + 1 ) − 1, (F1 , F2 ) is a distinguishable pair under the MM∗ model. Proof. Suppose (F1 , F2 ) is an indistinguishable pair. Assume, without loss of generality, that F2 − F1 = ∅. We will show that |F2 | > 2g (n − g + 1 ) − 1 so that we get a contradiction. By Lemma 4.4, we have F1 ∪ F2 = V(Hn ). Since F1 = F2 , then F1 F2 = ∅. By Lemma 4.9, Hn − F1 − F2 has no isolated vertex. Since F1 and F2 are indistinguishable, by Theorem 2.3(1), F1 ∩ F2 is a vertex cut of Hn . Since F1 and F2 are g-good-neighbor conditional faulty sets, F1 ∩ F2 is also a g-good-neighbor conditional faulty set which implies F1 ∩ F2 is a Rg -vertex-cut of Hn . By Theorem 3.3, |F1 ∩ F2 | ≥ 2g (n − g). Since F1 is a g-good-neighbor conditional faulty set, all the vertices in F2 − F1 have at least g neighbors out of F1 . By Theorem 2.3(1), EHn (F1 F2 , V (Hn ) − F1 − F2 ) = ∅, then δ (Hn [F2 − F1 ] ) ≥ g. Thus |F2 − F1 | ≥ 2g by Proposition 2.12. So
2g (n − g + 1 ) − 1 ≥ |F2 | = |F1 ∩ F2 | + |F2 − F1 | ≥ 2g (n − g) + 2g = 2g (n − g + 1 ), a contradiction.
By Theorem 4.2 and Lemma 4.10, we get the g-good neighbor of conditional diagnosability of Hn under the MM∗ model for 1 ≤ g ≤ n − 3.
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Theorem 4.11. For n ≥ 5 and 1 ≤ g ≤ n − 3, tg (Hn ) = 2g (n − g + 1 ) − 1 under the MM∗ model. For n − 2 ≤ g ≤ n − 1, we have the following result. Theorem 4.12. For n ≥ 5 and n − 2 ≤ g ≤ n − 1, tg (Hn ) = 2n−1 − 1 under the MM∗ model. Proof. By Proposition 2.6, tg (Hn ) ≥ tn−3 (Hn ). By Theorem 4.11, we have tn−3 (Hn ) = 2n−1 − 1, then tg (Hn ) ≥ 2n−1 − 1. Since tg (Hn ) ≤ 2n−1 − 1 by Theorem 4.3, then tg (Hn ) = 2n−1 − 1. 5. Conclusions In this paper, we consider the g-good-neighbor conditional diagnosability of the n-dimensional twisted hypercube Hn under the PMC model and MM∗ model, respectively. We show that when n ≥ 4, the g-good-neighbor conditional diagnosability of Hn under the PMC model is
tg (Hn ) =
2g ( n − g + 1 ) − 1 2n−1 − 1
if 0 ≤ g ≤ n − 3, if g − 2 ≤ g ≤ n − 1.
and when n ≥ 5, the g-good-neighbor conditional diagnosability of Hn under the MM∗ model is
tg (Hn ) =
2g ( n − g + 1 ) − 1 2n−1 − 1
if 0 ≤ g ≤ n − 3, if g − 2 ≤ g ≤ n − 1.
Observing that when g = 0, there is no restriction on the faulty sets and t0 (Hn ) = n is just the traditional diagnosability of Hn . However, when g = 1, t1 (Hn ) = 2n − 1. The difference between these two measures is that we only have restriction on the fault-free vertices instead of all vertices in the network. In the future, we may consider the diagnosability with the requirement of all the vertices having at least g good neighbors. Acknowledgments Huiqing Liu is partially supported by National Natural Science Foundation of China under grant numbers 11571096 and 61373019. Xiaolan Hu is partially supported by National Natural Science Foundation of China under grant number 11601176, Natural Science Foundation of Hubei Province under grant number 2016CFB146. References [1] G.Y. Chang, G.H. Chen, G.J. Chang, (t, k)-diagnosis for matching composition networks under the MM∗ model, IEEE Trans. Comput. 56 (1) (2007) 73–79. [2] N.W. Chang, S.Y. Hsieh, Structural properties and conditional diagnosability of star graphs by using PMC model, IEEE Trans. Parallel Distrib. Syst. 25 (11) (2014) 3002–3011. [3] C.A. Chen, S.Y. Hsieh, (t, k)-diagnosis for component-composition graphs under the MM∗ model, IEEE Trans. Comput. 60 (12) (2011) 1704–1717. [4] E. Cheng, K. Qiu, Z.Z. Shen, On the conditional diagnosability of matching composition networks, Theoret. Comput. Sci. 557 (2014) 101–114. [5] E. Cheng, K. Qiu, Z.Z. Shen, Diagnosability problems of the exchanged hypercube and its generalization, Int. J. Comput. Math Comput. Syst. Theor. 2 (2017) 39–52. [6] A.T. Dahbura, G.M. Masson, An o(n2.5 ) fault identification algorithm for diagnosable systems, IEEE Trans. Comput. 33 (1984) 486–492. [7] R.X. Hao, Y.Q. Feng, J.X. Zhou, Conditional diagnosability of alternating group graphs, IEEE Trans. Comput. 62 (4) (2013) 827–831. [8] R.X. Hao, Z.X. Tian, J.M. Xu, Relationship between conditional diagnosability and 2-extra connectivity of symmetric graphs, Theoret. Comput. Sci. 627 (2016) 36–53. [9] S.Y. Hsieh, C.W. Lee, Diagnosability of two-matching composition networks under the MM∗ model, IEEE Trans. Depend. Secure Comput. 8 (2) (2011) 246–255. [10] G.H. Hsu, C.F. Chiang, L.M. Shih, L.H. Hsu, J.J.M. Tan, Conditional diagnosability of hypercubes under the comparison diagnosis model, J. Syst. Archit. 55 (2) (2009) 140–146. [11] P.L. Lai, J.J.M. Tan, C.P. Chang, L.H. Hsu, Conditional diagnosability measures for large multiprocessor systems, IEEE Trans. Comput. 54 (2) (2005) 165–175. [12] S. Latifi, M. Hegde, M. Naraghi-Pour, Conditional connectivity measures for large multiprocessor systems, IEEE Trans. Comput. 43 (2) (1994) 218–222. [13] X.Y. Li, J.X. Fan, C.K. Lin, X.H. Jia, Diagnosability evaluation of the data center network DCell, Comput. J. 61 (1) (2018) 129–143. [14] D.S. Li, M. Lu, The g-good-neighbor conditional diagnosability of star graphs under the PMC and MM∗ model, Theoret. Comput. Sci. 674 (2017) 53–59. [15] X.J. Li, J.M. Xu, Edge-fault tolerance of hypercube-like networks, Inf. Process. Lett. 113 (2013) 760–763. [16] L. Lin, L. Xu, S. Zhou, Conditional diagnosability and strong diagnosability of split-star networks under the PMC model, Theoret. Comput. Sci. 562 (2015) 565–580. [17] S.L. Peng, C.K. Lin, J.J.M. Tan, L.H. Hsu, The g-good-neighbor conditional diagnosability of hypercube under PMC model, Appl. Math. Comput. 218 (21) (2012) 10406–10412. [18] F.P. Preparata, G. Metze, R.T. Chien, On the connection assignment problem of diagnosable systems, IEEE Trans. Electron. Comput. EC-16 (1967) 848–854. [19] A. Sengupta, A.T. Dahbura, On self-diagnosable multiprocessor systems: diagnosis by the comparison approach, IEEE Trans. Comput. 41 (11) (1992) 1386–1396. [20] A.S. Vaidya, P.S.N. Rao, S.R. Shankar, A class of hypercube-like networks, in: Proceedings of Fifth IEEE Symposium on Parallel and Distributed Processing, 1993, pp. 800–803. [21] M. Wang, Y. Guo, S. Wang, The 1-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM∗ model, Int. J. Comput. Math. 94 (3) (2016) 620–631. [22] S. Wang, W. Han, The g-good-neighbor conditional diagnosability of n-dimensional hypercubes under the MM∗ model, Inf. Process. Lett. 116 (2016) 574–577. [23] M. Wang, Y. Liu, S. Wang, The 2-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM∗ model, Theor. Comput. Sci. 628 (2016) 92–100.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The g-good neighbor conditional diagnosability of twisted hypercubes under the PMC and MM∗ model Huiqing Liu a, Xiaolan Hu b,a,∗, Shan Gao a a
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistic, Hubei University, Wuhan 430062, PR China School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, PR China
b
a r t i c l e
i n f o
Keywords: Twisted hypercubes g-good neighbor Rg -vertex-connectivity Conditional diagnosability PMC model MM∗ model
a b s t r a c t Connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks. The g-good-neighbor conditional faulty set is a special faulty set that every fault-free vertex should have at least g fault-free neighbors. The Rg -vertex-connectivity of a connected graph G is the minimum cardinality of a g-good-neighbor conditional faulty set X⊆V(G) such that G − X is disconnected. The g-good-neighbor conditional diagnosability is a metric that can give the maximum cardinality of g-good-neighbor conditional faulty set that the system is guaranteed to identify. The twisted hypercube is a new variant of hypercubes with asymptotically optimal diameter introduced by X.D. Zhu. In this paper, we first determine the Rg -vertex-connectivity of twisted hypercubes, then establish the g-good neighbor conditional diagnosability of twisted hypercubes under the PMC model and MM∗ model, respectively. © 2018 Elsevier Inc. All rights reserved.
1. Introduction A multiprocessor system comprises two or more processors, and various processors exchange information via links between them. With the rapid development of science and technology, the number of processors is increasing. As the size of multiprocessor systems increase, processor failure is inevitable. Therefore, the topic of reliability and fault-tolerance of multiprocessor systems has become an active area of intensive research. To evaluate the reliability of multiprocessor systems, fault diagnosability has become an important metric. There are several different diagnosis models being proposed to determine the diagnosability of a system. In this paper, we use the PMC model introduced by Preparata et al. [18] and MM∗ model suggested by Sengupta and Dahbura [19]. In the PMC model, every processor can test the processor that is adjacent to it. In the MM∗ model, every processor must test two processors if the processor is adjacent to the latter two processors. Many researchers have applied the PMC model and MM∗ model to identify faults in various topologies, see [1–3,9,16]. The diagnosability for multiprocessor systems assumes that all the neighbors of any processor may fail simultaneously. Therefore, for a system, its diagnosability is restricted by its minimum degree. However, the probability that this event occurs is very small in large-scale multiprocessor systems. To obtain a more practical diagnosability, Lai et al. [11] introduced
∗ Corresponding author at: School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, PR China. E-mail address: [email protected] (X. Hu).
https://doi.org/10.1016/j.amc.2018.03.042 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.
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conditional diagnosability under the assumption that all the neighbors of any processor in a multiprocessor system cannot be faulty at the same time. The conditional diagnosability of interconnection networks has been extensively investigated, see [2,4,7–10,25,30]. Recently, Peng et al. [17] proposed a new measure for fault diagnosis of the system, namely, g-good-neighbor conditional diagnosability, which requires that every fault-free node contains at least g fault-free neighbors. Compared with diagnosability, the g-good-neighbor conditional diagnosability improves accuracy in measuring the reliability of interconnection networks in heterogeneous environments. Meanwhile, Peng et al. [17] studied the g-good-neighbor conditional diagnosability of the n-dimensional hypercube Qn under the PMC model. Wang and Han [22] determined the g-good-neighbor diagnosability of the n-dimensional hypercube Qn under the MM∗ model. Since then, numerous studies have been investigated under the PMC model and MM∗ model, see [5,13,14,21,23,24,27,28]. In this paper, we consider the g-good-neighbor conditional diagnosability of the n-dimensional twisted hypercube Hn , which is a new variant of hypercubes with asymptotically optimal diameter introduced by Zhu [29], under the PMC model and MM∗ model, respectively. Our main results are listed below. Theorem 1.1. Let n be an integer with n ≥ 4. Then the g-good-neighbor conditional diagnosability of Hn under the PMC model is
tg (Hn ) =
2g ( n − g + 1 ) − 1 2n−1 − 1
if 0 ≤ g ≤ n − 3, if g − 2 ≤ g ≤ n − 1.
Theorem 1.2. Let n be an integer with n ≥ 5. Then the g-good-neighbor conditional diagnosability of Hn under the MM∗ model is
tg (Hn ) =
2g ( n − g + 1 ) − 1 2n−1 − 1
if 0 ≤ g ≤ n − 3, if g − 2 ≤ g ≤ n − 1.
The rest of this paper is organized as follows: Section 2 provides preliminaries for our notations, diagnosing a system and the twisted hypercubes. In Section 3, we determine the Rg -vertex-connectivity of twisted hypercubes. In Sections 4, we establish the g-good-neighbor conditional diagnosability of twisted hypercubes under the PMC model and MM∗ model, respectively. Our conclusions are given in Sections 5. 2. Preliminaries 2.1. Notations Let G = (V (G ), E (G )) be a simple and finite graph. The neighborhood NG (v ) of a vertex v is the set of vertices adjacent to v and the closed neighborhood of v is NG [v] = NG (v ) ∪ {v}. The degree dG (v ) of v is |NG (v )|. The minimum degree of G is denoted by δ (G). If dG (v ) = k for any v ∈ V (G ), then G is called a k-regular graph. For S⊆V(G), G[S] denotes the subgraph induced by S. The neighborhood set of S is defined as NG (S ) = (∪v∈S NG (v )) − S, and the closed neighborhood set of S is defined as NG [S] = NG (S ) ∪ S. We will use G − S to denote the subgraph G[V (G ) − S]. For any v ∈ V (G ), NS (v ) denotes the neighborhood of v in S. For two disjoint subsets S, T of V(G), let EG (S, T ) = {uv ∈ E (G ) | u ∈ S, v ∈ T }. The symmetric difference of F1 ⊆V(G) and F2 ⊆V(G) is defined as the set F1 F2 = (F1 − F2 ) ∪ (F2 − F1 ). Let Kn be a complete graph of order n. The minimum cardinality of a vertex set S⊆V(G) such that G − S is disconnected or has only one vertex, denoted by κ (G), is the connectivity of G. A subset F⊆V(G) is called a Rg -vertex-set of G if δ (G − F ) ≥ g. A Rg -vertex-cut of a connected graph G is a Rg -vertex-set F such that G − F is disconnected. The Rg -vertex-connectivity of G, denoted by κ g (G), is the cardinality of a minimum Rg -vertex-cut of G. Note that κ 0 (G ) = κ (G ). 2.2. The PMC model and MM∗ model for diagnosis A multiprocessor system is typically represented by an undirected simple graph G = (V, E ), where V(G) stands for the processors and E(G) represents the links between two processors. Preparata et al. [18] proposed the PMC model, which performs diagnosis by testing the neighboring processor via the links between them. Under the PMC model, tests can be performed between any two adjacent vertices u and v. We use the ordered pair (u, v ) to denote a test that u diagnoses v. The result of a test (u, v ) is reliable if and only if u is fault-free. In this condition, the result is 0 if v is fault-free, and is 1 otherwise. A test assignment for a system G is a collection of tests, which can be represented a directed graph T = (V (G ), L ), where V(G) is the vertex set of G and L = {(u, v ) | u, v ∈ V (G ) and uv ∈ E (G )}. The collection of all test results from the test scheme T is termed as a syndrome σ : L → {0, 1}. Let T = (V (G ), L ) be a test assignment, and F a subset of V(G). For any given syndrome σ resulting from T, F is said to be consistent with σ if the syndrome σ can be produced when all vertices in F are faulty and all vertices in V (G ) − F are fault-free. That is, if u is fault-free, then σ ((u, v )) = 0 if v is fault-free, and σ ((u, v )) = 1 if v is faulty; if u is faulty, then σ ((u, v )) can be either 0 or 1 no matter v is faulty or not. Therefore, a faulty set F may be consistent with different syndromes. We use σ (F) to represent the set of all possible syndromes with which the faulty set F can be consistent.
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Fig. 1. Illustration of a distinguishable pair under the PMC model.
Fig. 2. Illustration of a distinguishable pair under the MM∗ model.
Let F1 and F2 be two distinct faulty sets of V(G), F1 and F2 are distinguishable if σ (F1 ) ∩ σ (F2 ) = ∅; otherwise, F1 and F2 are indistinguishable. In other words, if σ (F1 ) ∩ σ (F2 ) = ∅, then (F1 , F2 ) is a distinguishable pair; otherwise, (F1 , F2 ) is an indistinguishable pair. A system G is called t-diagnosable, if any two distinct faulty sets F1 , F2 ⊆V(G) are distinguishable, provided that |F1 |, |F2 | ≤ t. The diagnosability of G, denoted by t(G), is the maximum t such that G is t-diagnosable. In [6], Dahbura and Masson proposed a sufficient and necessary condition of t-diagnosable systems. Theorem 2.1. [6] A system G = (V, E ) is t-diagnosable if and only if, for any two distinct subsets F1 and F2 of V(G) with |F1 |, |F2 | ≤ t, there is at least one test from V (G ) − F1 − F2 to F1 F2 . For the PMC model, Peng et al. [17] proposed the following theorem. Theorem 2.2. [17] For any two distinct subsets F1 and F2 of V(G), (F1 , F2 ) is a distinguishable pair under the PMC model if and only if there is a vertex u ∈ V (G ) − F1 − F2 and a vertex v ∈ F1 F2 such that uv ∈ E (G ) (see Fig. 1). Sengupta and Dahbura [19] proposed the MM∗ model, which performs diagnosis by sending the same inputs to a pair of adjacent processors, and comparing their responses. In the MM∗ model, tests can be performed from any vertex w to its any two neighbors u and v. We use the labeled pair (u, v )w to denote a test performed by w on its neighbors u and v. The result of a test (u, v )w is reliable if and only if w is fault-free. In this condition, the result is 0 if both u and v are fault-free, and is 1 otherwise. The test scheme of a system G is often represented by a multigraph M = (V (G ), L ), where L = {(u, v )w | u, v, w ∈ V (G ) and uw, vw ∈ E (G )}. Since a pair of vertices may be compared by different vertices, M is a multigraph. The collection of all test results from the test scheme M is termed as a syndrome σ : L → {0, 1}. Given a subset of vertices F⊆V(G), we say that F is consistent with σ if the syndrome σ can be produced when all nodes in F are faulty and all nodes in V (G ) − F are fault-free. That is, if w is fault-free, then σ ((u, v )w ) = 0 if u and v are fault-free, and σ ((u, v )w ) = 1 otherwise; if w is faulty, then σ ((u, v )w ) can be either 0 or 1 no matter u and v are faulty or not. Sengupta and Dahbura [19] proposed a sufficient and necessary condition for two distinct subsets F1 and F2 to be a distinguishable pair under the MM∗ model. Theorem 2.3. [19] For any two distinct subsets F1 and F2 of V(G), (F1 , F2 ) is a distinguishable pair under the MM∗ model if and only if one of the following conditions is satisfied (see Fig. 2): (1) There are two vertices u, w ∈ V (G ) − F1 − F2 and there is a vertex v ∈ F1 F2 such that uw, vw ∈ E (G ), (2) There are two vertices u, v ∈ F1 − F2 and there is a vertex w ∈ V (G ) − F1 − F2 such that uw, vw ∈ E (G ), (3) There are two vertices u, v ∈ F2 − F1 and there is a vertex w ∈ V (G ) − F1 − F2 such that uw, vw ∈ E (G ). In the following, we will give the definition of g-good-neighbor conditional faulty set and the g-good-neighbor conditional diagnosability of a system. Definition 2.4. Let G = (V, E ) be a system. A faulty set F⊆V(G) is called a g-good-neighbor conditional faulty set if δ (G − F ) ≥ g. Definition 2.5. A system G = (V, E ) is g-good-neighbor conditional t-diagnosable, if any two distinct g-good-neighbor conditional faulty sets F1 , F2 ⊆V(G) are distinguishable, provided that |F1 |, |F2 | ≤ t. The g-good-neighbor conditional diagnosability of G, denoted by tg (G), is the maximum t such that G is g-good-neighbor conditionally t-diagnosable. Note that t0 (g) = t (G ). By definition of the g-good-neighbor conditional diagnosability, we have the following proposition which is also observed in [17].
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Fig. 3. H3 and H4 .
Proposition 2.6. [17] For any system G = (V, E ), if g ≤ g , then tg (G ) ≤ tg (G ) under the PMC model and MM∗ model, respectively. 2.3. The twisted hypercubes In this subsection, we first introduce the definition of the n-dimensional twisted hypercube Hn , and then give some properties of Hn . Denote by Z2n the set of binary strings of length n. For x, y ∈ Z2n , xy denotes the sum of x and y in the group Z2n , i.e., (x y )i = xi + yi (mod 2 ) (for x ∈ Z2n , xi denotes the ith bit of x). If x is a binary string of length n1 and y is a binary string of length n2 , then xy is the concatenation of x and y, which is a binary string of length n1 + n2 . If Z is a set of binary strings, then let xZ = {xy : y ∈ Z }. For x ∈ Z2n , we use x[i, j] to denote the binary string xi xi+1 . . . x j for 1 ≤ i < j ≤ n. We first present an integer function κ (n). Definition 2.7. Let κ be the integer function defined as the following:
κ (n ) =
0 max{1, log2 n − 2 log2 log2 n}
if n = 1, otherwise.
Next, a permutation φ of binary strings is given as follows. Definition 2.8. Assume x ∈ Z2n and n ≥ 2. Then φ (x ) ∈ Z2n is the binary string such that φ (x )[1, κ (n )] = x[1, κ (n )] x[n − κ (n ) + 1, n], φ ( x )[ κ ( n ) + 1 , n ] = x [ κ ( n ) + 1 , n ] . Note that the restriction of φ to Z2n is indeed a permutation of Z2n , with φ 2 (x ) = x. Now we give a recursive definition of the n-dimensional twisted hypercube Hn . Definition 2.9. Set H1 := K2 , with vertices 0 and 1. For n ≥ 2, Hn is obtained from two copies of Hn−1 , 0Hn−1 and 1Hn−1 , by adding edges connecting 0x and 1φ (x) for all x ∈ V (Hn−1 ). The vertex set of Hn is Z2n . It follows from the definition that H1 = K2 , H2 = C4 , H3 and H4 are depicted in Fig. 3. It is seen that Hn is a kind of n-dimensional hypercube-like networks. Hypercube-like network was first proposed by Vaidya et al. [20] for a unified study on hypercube variants. By the definition of Hn , we have the following result which is also observed in [29]. q
n−q
Proposition 2.10. For 1 ≤ q < n and b ∈ Z2 , the subgraph induced by bZ2
is a Hn−q .
Let a ∈ {0, 1} be an integer and a the complement of a, i.e., a = 1 − a. Let v = v1 v2 · · · vn be any vertex of Hn . Then v q has only one neighbor φ (v ) in v1 Hn−1 , and n − 1 neighbors in v1 Hn−1 . By Proposition 2.10, for 1 ≤ q ≤ n − 1 and b ∈ Z2 , the n−q subgraph induced by bZ2 is a Hn−q . So for every neighbor w = w1 w2 · · · wn of v in v1 Hn−1 , there exists some q (1 ≤ q ≤ n − 1 ) such that wi = vi for 1 ≤ i ≤ q. The following results will be used in Sections 3 and 4. Proposition 2.11. For any u, v ∈ V (Hn ), u and v have at most two common neighbors. Proof. We prove this proposition by induction on n. In the basis step, for n = 2, H2 is a 4-cycle. It is seen that any two vertices of H2 have at most two common neighbors. In the induction step, assume that the statement is true for n = k − 1. That is, any two vertices of Hk−1 have at most two common neighbors. When n = k, let u and v be two vertices of Hk .
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If u, v ∈ V (0Hk−1 ) or u, v ∈ V (1Hk−1 ), say u, v ∈ V (0Hk−1 ), then u and v have at most two common neighbors in 0Hk−1 by induction hypothesis. Note that u and v have no common neighbors in 1Hk−1 . Then u and v have at most two common neighbors. Now we assume u ∈ V (0Hk−1 ) and v ∈ V (1Hk−1 ). If v = φ (u ), then u and v have no common neighbors. So we can assume v = φ (u ). Then the common neighbor(s) of u and v will be φ (u) and/or φ −1 (v ). Therefore u and v have at most two common neighbors. Proposition 2.12. [15] Let Q be a subgraph of Hn . If δ (Q) ≥ g, then |V(Q)| ≥ 2g . For any subgraph Q of Hn , we use NHn [Q] to denote the closed neighborhood set of V(Q) in Hn . Proposition 2.13. [26] Let Q be a subgraph of Hn . If δ (Q) ≥ g, then |NHn [Q]| ≥ 2g (n − g + 1 ). For convenience, we write the consecutive k 0’s in the binary string as 0k . Proposition 2.14. Let X = {v1 v2 · · · vg 0n−g | vi ∈ {0, 1}, 1 ≤ i ≤ g} ⊂ V (Hn ) and F1 = NHn (X ). Then Hn [X] is a g-regular graph, and |F1 | = 2g (n − g). Proof. Let v = v1 v2 · · · vg 0n−g be any vertex of X, then NF1 (v ) = {v1 v2 · · · vg 0n−g− j−1 10 j | 0 ≤ j ≤ n − g − 1}. Thus F1 = n−g− j−1 10 j | v ∈ {0, 1}, 1 ≤ i ≤ g}. It is seen that v has g neighbors in X and n − g neighbors in F . 1 i 0≤ j≤n−g−1 {v1 v2 · · · vg 0 Then Hn [X] is a g-regular graph. Note that |X | = 2g , and any two different vertices u, v ∈ X have no common neighbors in F1 . Then |F1 | = 2g (n − g). 3. The Rg -vertex-connectivity of Hn In order to obtain the g-good-neighbor conditional diagnosability of Hn , we have to investigate the Rg -vertex-connectivity of Hn , which is closely related to g-good-neighbor conditional diagnosability proposed by Latifi et al. [12]. As a more refined index than the traditional connectivity, the Rg -vertex-connectivity can be used to measure the conditional fault tolerance of networks. In this section, we determine the Rg -vertex-connectivity of Hn for 0 ≤ g ≤ n − 2. In [26], Ye and Liang considered the Rg -vertex-connectivity of any n-dimensional hypercube-like network and got the following lower bound. Lemma 3.1. [26] For 0 ≤ g ≤ n − 2, κ g (Hn ) ≥ 2g (n − g). Lemma 3.2. For 0 ≤ g ≤ n − 2, let X = {v1 v2 · · · vg 0n−g | vi ∈ {0, 1}, 1 ≤ i ≤ g} ⊂ V(Hn ), F1 = NHn (X ) and F2 = NHn [X]. Then F1 is a g-good-neighbor conditional faulty set and F2 is a max{g, n − 3}-good-neighbor conditional faulty set of Hn . Proof. By Proposition 2.14, Hn [X] is a g-regular graph. So we only need to show that each vertex in V (Hn ) − F2 has at least max{g, n − 3} neighbors out of F2 , that is, each vertex in V (Hn ) − F2 has at most min{n − g, 3} neighbors in F1 . Let u = u1 u2 · · · un−1 un be a vertex in V (Hn ) − F2 . Note that F1 = 0≤ j≤n−g−1 {v1 v2 · · · vg 0n−g− j−1 10 j | vi ∈ {0, 1}, 1 ≤ i ≤ g} ⊂ F2 , then u has t ≥ 2 bits equal 1 between the last n − g bits. That is, there exists t ≥ 2 such that uki = 1 and g + 1 ≤ ki ≤ n for any 1 ≤ i ≤ t. Assume that g + 1 ≤ k1 < · · · < kt ≤ n. If g = n − 2, then u = u1 u2 · · · un−2 11, thus u has 2 = n − g neighbors {u1 u2 · · · un−2 10, u1 u2 · · · un−2 01} in F1 . So we assume g ≤ n − 3. Let v = u1 u2 · · · ug 0n−g be a vertex in X. If all the neighbors of u in F1 have the same first g bits as that of u, then NF1 (u ) ⊆ {u1 u2 · · · ug 0n−g− j−1 10 j | 0 ≤ j ≤ n − g − 1} = NF1 (v ). By Proposition 2.11, u and v have at most two common neighbors, then u has at most 2 < n − g neighbors in F1 . So we assume u has a neighbor w = w1 w2 · · · wn ∈ F1 such that w has the same first q (0 ≤ q ≤ g − 1) bits as that of u. Then w = u1 u2 · · · uq uq+1 (uq+2 · · · uq+κ +1 un−κ +1 · · · un )uq+κ +2 · · · un , where κ = κ (n − q − 1 ). Note that w ∈ F1 , then for any 1 ≤ i ≤ t − 1, wki = uki un−q−κ +ki −1 = 0, thus un−q−κ +ki −1 = 1. Since n − q − 1 ≥ n − g ≥ 3 and n − q − 1 ≥ 2 log2 (n − q − 1 ) − log2 log2 (n − q − 1 ), then n − q − 1 ≥ 2κ , i.e., n − κ + 1 ≥ q + κ + 2. Note that ki ≥ g + 1 ≥ q + 2, then n − q − κ + ki − 1 ≥ n − κ + 1, thus un−q−κ +ki −1 = wn−q−κ +ki −1 for 1 ≤ i ≤ t − 1, hence t = 2. If wk2 = uk2 un−q−κ +k2 −1 = 0, then un−q−κ +k2 −1 = 1 and un−q−κ +k2 −1 = wn−q−κ +k2 −1 , a contradiction with w ∈ F1 . Therefore wk2 = 1 and n − q − κ + k1 − 1 = k2 . Hence u has only one neighbor of w such that w has the same first q (q is an integer satisfying (n − q − 1 ) − κ (n − q − 1 ) = k2 − k1 ) bits as that of u, all the other neighbors of u are in NF1 (v ). By Proposition 2.11, u has at most two neighbors in NF1 (v ). Thus u has at most 3 ≤ n − g neighbors in F1 . Therefore we complete the proof of Lemma 3.2. Theorem 3.3. For 0 ≤ g ≤ n − 2, κ g (Hn ) = 2g (n − g). Proof. Let X = {v1 v2 · · · vg 0n−g | vi ∈ {0, 1}, 1 ≤ i ≤ g} ⊂ V(Hn ) and F1 = NHn (X ). Clearly, Hn − F1 is disconnected. By Lemma 3.2, F1 is a g-good-neighbor conditional faulty set. Then F1 is a Rg -vertex-cut of Hn . By Proposition 2.14, |F1 | = 2g (n − g). Then κ g (Hn ) ≤ 2g (n − g). By Lemma 3.1, κ g (Hn ) ≥ 2g (n − g). Hence κ g (Hn ) = 2g (n − g). 4. The g-good-neighbor conditional diagnosability of Hn In this section, we will give some lemmas, then determine tg (Hn ) for 0 ≤ g ≤ n − 1 under the PMC model and MM∗ model, respectively.
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Lemma 4.1. For 0 ≤ g ≤ n − 2, let X = {v1 v2 · · · vg 0n−g | vi ∈ {0, 1}, 1 ≤ i ≤ g} ⊂ V(Hn ), F1 = NHn (X ) and F2 = NHn [X]. Then (F1 , F2 ) is an indistinguishable pair under the PMC model and MM∗ model, respectively. Proof. Note that F1 F2 = X and NHn (X ) = F1 . Then (F1 , F2 ) is an indistinguishable pair under the PMC model by Theorem 2.2, and under the MM∗ model by Theorem 2.3. From Propositions 2.12, 2.13, Lemmas 3.2, 4.1, and combining with the definition of tg , we have the following result. Theorem 4.2. For 0 ≤ g ≤ n − 2, tg (Hn ) ≤ 2g (n − g + 1 ) − 1 under the PMC model and MM∗ model, respectively. For 0 ≤ g ≤ n − 1, tg (Hn ) has the following upper bound. Theorem 4.3. For 0 ≤ g ≤ n − 1, tg (Hn ) ≤ 2n−1 − 1 under the PMC model and MM∗ model, respectively. Proof. Let F1 = V (0Hn−1 ) and F2 = V (1Hn−1 ). Then |F1 | = |F2 | = 2n−1 . Note that F1 and F2 are g-good-neighbor conditional faulty sets. Since F1 ∪ F2 = V (Hn ), then (F1 , F2 ) is an indistinguishable pair under the PMC model by Theorem 2.2, and under the MM∗ model by Theorem 2.3. Thus tg (Hn ) ≤ 2n−1 − 1. Lemma 4.4. Let F1 and F2 be two distinct g-good-neighbor conditional faulty sets in Hn such that |F1 |, |F2 | ≤ 2g (n − g + 1 ) − 1. For 0 ≤ g ≤ n − 3, we have F1 ∪ F2 = V(Hn ). Proof. Since n − g ≥ 3, we have 2n−g−1 ≥ n − g + 1. Then
|F1 ∪ F2 | ≤ |F1 | + |F2 | ≤ 2g+1 (n − g + 1 ) − 2 ≤ 2g+1 2n−g−1 − 2 < 2n . Thus F1 ∪ F2 = V(Hn ).
4.1. tg (Hn ) under the PMC model In this subsection, we will determine tg (Hn ) for n ≥ 4 and 0 ≤ g ≤ n − 1 under the PMC model. Lemma 4.5. Let 0 ≤ g ≤ n − 3. For any two distinct g-good-neighbor conditional faulty sets F1 and F2 in Hn with |F1 |, |F2 | ≤ 2g (n − g + 1 ) − 1, (F1 , F2 ) is a distinguishable pair under the PMC model. Proof. Suppose that (F1 , F2 ) is an indistinguishable pair. Assume, without loss of generality, that F2 − F1 = ∅. By Lemma 4.4, we have F1 ∪ F2 = V(Hn ). Since F1 = F2 , then F1 F2 = ∅. By Theorem 2.2, F1 ∩ F2 is a vertex cut of Hn . Since F1 and F2 are g-good-neighbor conditional faulty sets of Hn , F1 ∩ F2 is also a g-good-neighbor conditional faulty set which implies F1 ∩ F2 is a Rg -vertex-cut of Hn . By Theorem 3.3, |F1 ∩ F2 | ≥ 2g (n − g). Since F1 is a g-good-neighbor conditional faulty set, all the vertices in F2 − F1 have at least g neighbors out of F1 . By Theorem 2.2, EHn (F1 F2 , V (Hn ) − F1 − F2 ) = ∅, then δ (Hn [F2 − F1 ] ) ≥ g. Thus |F2 − F1 | ≥ 2g by Proposition 2.12. So
2g (n − g + 1 ) − 1 ≥ |F2 | = |F1 ∩ F2 | + |F2 − F1 | ≥ 2g (n − g) + 2g = 2g (n − g + 1 ), a contradiction.
By Theorem 4.2 and Lemma 4.5, we get the g-good neighbor of conditional diagnosability of Hn under the PMC model for n ≥ 4 and 0 ≤ g ≤ n − 3. Theorem 4.6. For n ≥ 4 and 0 ≤ g ≤ n − 3, tg (Hn ) = 2g (n − g + 1 ) − 1 under the PMC model. For n − 2 ≤ g ≤ n − 1, we have the following result. Theorem 4.7. For n ≥ 4 and n − 2 ≤ g ≤ n − 1, tg (Hn ) = 2n−1 − 1 under the PMC model. Proof. By Proposition 2.6, tg (Hn ) ≥ tn−3 (Hn ). Note that tn−3 (Hn ) = 2n−1 − 1 by Theorem 4.6, then tg (Hn ) ≥ 2n−1 − 1. Since tg (Hn ) ≤ 2n−1 − 1 by Theorem 4.3, then tg (Hn ) = 2n−1 − 1. 4.2. tg (Hn ) under the MM∗ model In this subsection, we will determine tg (Hn ) for n ≥ 5 and 0 ≤ g ≤ n − 1 under the MM∗ model. First we consider the case of g = 0, the 0-good-neighbor condition does not have any restriction on the faulty sets in this case, then t0 (Hn ) = t (Hn ). Theorem 4.8. For n ≥ 5, t0 (Hn ) = n under the MM∗ model. Proof. First we show that t0 (Hn ) ≤ n. Let v ∈ V (Hn ), F1 = NHn (v ) and F2 = NHn [v]. Note that F1 F2 = {v}, then (F1 , F2 ) is an indistinguishable pair by Theorem 2.3, thus t0 (Hn ) ≤ n by the definition of t0 . Now we show that t0 (Hn ) ≥ n. We need to show that for any two distinct faulty sets F1 , F2 in Hn with |F1 |, |F2 | ≤ n, (F1 , F2 ) is a distinguishable pair. Suppose to the contrary that (F1 , F2 ) is an indistinguishable pair. Assume, without loss of generality,
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that F2 − F1 = ∅, then |F2 − F1 | > 0. Note that |F1 ∪ F2 | ≤ |F1 | + |F2 | ≤ 2n < 2n for n ≥ 5, then F1 ∪ F2 = V(Hn ). Since F1 = F2 , then F1 F2 = ∅. By Theorem 2.1, F1 ∩ F2 is a vertex cut of Hn . By Theorem 3.3, |F1 ∩ F2 | ≥ n. So
n ≥ |F2 | = |F1 ∩ F2 | + |F2 − F1 | > |F1 ∩ F2 | ≥ n, a contradiction. Therefore, we complete the proof of Theorem 4.8.
In the following, we assume that 1 ≤ g ≤ n − 1. We will determine tg (Hn ) for 1 ≤ g ≤ n − 3 and g − 2 ≤ g ≤ n − 1, respectively. Lemma 4.9. For n ≥ 5 and 1 ≤ g ≤ n − 2. Let F1 and F2 be two distinct g-good-neighbor conditional faulty sets such that |F1 |, |F2 | ≤ 2g (n − g + 1 ) − 1. If F1 and F2 are indistinguishable under the MM∗ model, then Hn − F1 − F2 has no isolated vertex. Proof. Let W be the set of isolated vertices in Hn − F1 − F2 , and Y the subgraph induced by the vertex set V (Hn ) − F1 − F2 − W . We will show that W = ∅. Suppose to the contrary that W = ∅ and let w ∈ W . We consider the following two cases. Case 1. g = 1. In this case, |F1 |, |F2 | ≤ 2n − 1. Since F1 is a 1-good-neighbor conditional faulty set, w must have at least one neighbor out of F1 . Note that w is an isolated vertex of Hn − F1 − F2 , then NHn −F1 (w ) ⊆ F2 − F1 . Since F1 and F2 are indistinguishable, then |NHn −F1 (w )| = 1 by Theorem 2.3(3). Similarly, NHn −F2 (w ) ⊆ F1 − F2 and |NHn −F2 (w )| = 1. Therefore |NF1 ∩F2 (w )| = n − 2. So we have
|NF1 ∩F2 (w )| = (n − 2 )|W | ≤
w∈W
dHn (v ) = n|F1 ∩ F2 | ≤ n(|F1 | − 1 ) ≤ n(2n − 2 ).
v∈F1 ∩F2
Thus |W | ≤ n(2n − 2 )/(n − 2 ) ≤ 2n + 4 as n ≥ 5. If V (Y ) = ∅, then |V (Hn )| = |F1 ∪ F2 | + |W |. Note that n ≥ 5, then
2(2n − 1 ) ≥ |F1 | + |F2 | = |F1 ∪ F2 | + |F1 ∩ F2 | = |V (Hn )| − |W | + |F1 ∩ F2 | ≥ 2n − (2n + 4 ) + (n − 2 ) > 4n, a contradiction. So V(Y) = ∅. Since F1 and F2 are indistinguishable, F1 ∩ F2 is a R1 -vertex-cut of Hn by Theorem 2.3. By Theorem 3.3, |F1 ∩ F2 | ≥ 2n − 2. Note that for any w ∈ W, w has one neighbor in F1 − F2 and one neighbor in F2 − F1 . So we have |F1 ∪ F2 | = 2n − 2 which implies |F1 | = |F2 | = 2n − 1 and |F1 − F2 | = |F2 − F1 | = 1. Assume F1 − F2 = {v1 } and F2 − F1 = {v2 }. For any w ∈ W, we have v1 , v2 ∈ NHn (w ). By Proposition 2.11, |W| ≤ 2. By definition of Hn , Hn has no triangles, then w and vi have no common neighbors for i = 1, 2. If |W | = 1, then W = {w} and |NF1 ∪F2 (w )| = n − 2, |NF1 ∪F2 (v1 )| = n − 1 and |NF1 ∪F2 (v2 )| = n − 1. By Proposition 2.11, v1 and v2 have at most one common neighbor in F1 ∪ F2 . So
2n − 2 = |F1 ∪ F2 | ≥ (n − 2 ) + (n − 1 ) + (n − 2 ) = 3n − 5, i.e., n ≤ 3, a contradiction with n ≥ 5. If |W | = 2, denote W = {w1 , w2 }. Then |NF1 ∪F2 (wi )| = n − 2 for 1 ≤ i ≤ 2, |NF1 ∪F2 (v1 )| = n − 2 and |NF1 ∪F2 (v2 )| = n − 2. By Proposition 2.11, v1 and v2 have no common neighbors in F1 ∪ F2 , w1 and w2 have no common neighbors in F1 ∪ F2 . So
2n − 2 = |F1 ∪ F2 | ≥ (n − 2 ) + (n − 2 ) + (n − 2 ) + (n − 2 ) = 4n − 8, i.e., n ≤ 3, a contradiction with n ≥ 5. Case 2. g ≥ 2. Since F1 is a g-good-neighbor conditional faulty set, then |NHn −F1 (w )| ≥ g. Note that w is an isolated vertex in Hn − F1 − F2 , then NHn −F1 (w ) ⊆ F2 − F1 . Since g ≥ 2, by Theorem 2.3(3), F1 and F2 are distinguishable, a contradiction. Lemma 4.10. Let n ≥ 5 and 1 ≤ g ≤ n − 3. For any two distinct g-good-neighbor conditional faulty sets F1 and F2 with |F1 |, |F2 | ≤ 2g (n − g + 1 ) − 1, (F1 , F2 ) is a distinguishable pair under the MM∗ model. Proof. Suppose (F1 , F2 ) is an indistinguishable pair. Assume, without loss of generality, that F2 − F1 = ∅. We will show that |F2 | > 2g (n − g + 1 ) − 1 so that we get a contradiction. By Lemma 4.4, we have F1 ∪ F2 = V(Hn ). Since F1 = F2 , then F1 F2 = ∅. By Lemma 4.9, Hn − F1 − F2 has no isolated vertex. Since F1 and F2 are indistinguishable, by Theorem 2.3(1), F1 ∩ F2 is a vertex cut of Hn . Since F1 and F2 are g-good-neighbor conditional faulty sets, F1 ∩ F2 is also a g-good-neighbor conditional faulty set which implies F1 ∩ F2 is a Rg -vertex-cut of Hn . By Theorem 3.3, |F1 ∩ F2 | ≥ 2g (n − g). Since F1 is a g-good-neighbor conditional faulty set, all the vertices in F2 − F1 have at least g neighbors out of F1 . By Theorem 2.3(1), EHn (F1 F2 , V (Hn ) − F1 − F2 ) = ∅, then δ (Hn [F2 − F1 ] ) ≥ g. Thus |F2 − F1 | ≥ 2g by Proposition 2.12. So
2g (n − g + 1 ) − 1 ≥ |F2 | = |F1 ∩ F2 | + |F2 − F1 | ≥ 2g (n − g) + 2g = 2g (n − g + 1 ), a contradiction.
By Theorem 4.2 and Lemma 4.10, we get the g-good neighbor of conditional diagnosability of Hn under the MM∗ model for 1 ≤ g ≤ n − 3.
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Theorem 4.11. For n ≥ 5 and 1 ≤ g ≤ n − 3, tg (Hn ) = 2g (n − g + 1 ) − 1 under the MM∗ model. For n − 2 ≤ g ≤ n − 1, we have the following result. Theorem 4.12. For n ≥ 5 and n − 2 ≤ g ≤ n − 1, tg (Hn ) = 2n−1 − 1 under the MM∗ model. Proof. By Proposition 2.6, tg (Hn ) ≥ tn−3 (Hn ). By Theorem 4.11, we have tn−3 (Hn ) = 2n−1 − 1, then tg (Hn ) ≥ 2n−1 − 1. Since tg (Hn ) ≤ 2n−1 − 1 by Theorem 4.3, then tg (Hn ) = 2n−1 − 1. 5. Conclusions In this paper, we consider the g-good-neighbor conditional diagnosability of the n-dimensional twisted hypercube Hn under the PMC model and MM∗ model, respectively. We show that when n ≥ 4, the g-good-neighbor conditional diagnosability of Hn under the PMC model is
tg (Hn ) =
2g ( n − g + 1 ) − 1 2n−1 − 1
if 0 ≤ g ≤ n − 3, if g − 2 ≤ g ≤ n − 1.
and when n ≥ 5, the g-good-neighbor conditional diagnosability of Hn under the MM∗ model is
tg (Hn ) =
2g ( n − g + 1 ) − 1 2n−1 − 1
if 0 ≤ g ≤ n − 3, if g − 2 ≤ g ≤ n − 1.
Observing that when g = 0, there is no restriction on the faulty sets and t0 (Hn ) = n is just the traditional diagnosability of Hn . However, when g = 1, t1 (Hn ) = 2n − 1. The difference between these two measures is that we only have restriction on the fault-free vertices instead of all vertices in the network. In the future, we may consider the diagnosability with the requirement of all the vertices having at least g good neighbors. Acknowledgments Huiqing Liu is partially supported by National Natural Science Foundation of China under grant numbers 11571096 and 61373019. Xiaolan Hu is partially supported by National Natural Science Foundation of China under grant number 11601176, Natural Science Foundation of Hubei Province under grant number 2016CFB146. References [1] G.Y. Chang, G.H. Chen, G.J. Chang, (t, k)-diagnosis for matching composition networks under the MM∗ model, IEEE Trans. Comput. 56 (1) (2007) 73–79. [2] N.W. Chang, S.Y. Hsieh, Structural properties and conditional diagnosability of star graphs by using PMC model, IEEE Trans. Parallel Distrib. Syst. 25 (11) (2014) 3002–3011. [3] C.A. Chen, S.Y. Hsieh, (t, k)-diagnosis for component-composition graphs under the MM∗ model, IEEE Trans. Comput. 60 (12) (2011) 1704–1717. [4] E. Cheng, K. Qiu, Z.Z. Shen, On the conditional diagnosability of matching composition networks, Theoret. Comput. Sci. 557 (2014) 101–114. [5] E. Cheng, K. Qiu, Z.Z. Shen, Diagnosability problems of the exchanged hypercube and its generalization, Int. J. Comput. Math Comput. Syst. Theor. 2 (2017) 39–52. [6] A.T. Dahbura, G.M. Masson, An o(n2.5 ) fault identification algorithm for diagnosable systems, IEEE Trans. Comput. 33 (1984) 486–492. [7] R.X. Hao, Y.Q. Feng, J.X. Zhou, Conditional diagnosability of alternating group graphs, IEEE Trans. Comput. 62 (4) (2013) 827–831. [8] R.X. Hao, Z.X. Tian, J.M. Xu, Relationship between conditional diagnosability and 2-extra connectivity of symmetric graphs, Theoret. Comput. Sci. 627 (2016) 36–53. [9] S.Y. Hsieh, C.W. Lee, Diagnosability of two-matching composition networks under the MM∗ model, IEEE Trans. Depend. Secure Comput. 8 (2) (2011) 246–255. [10] G.H. Hsu, C.F. Chiang, L.M. Shih, L.H. Hsu, J.J.M. Tan, Conditional diagnosability of hypercubes under the comparison diagnosis model, J. Syst. Archit. 55 (2) (2009) 140–146. [11] P.L. Lai, J.J.M. Tan, C.P. Chang, L.H. Hsu, Conditional diagnosability measures for large multiprocessor systems, IEEE Trans. Comput. 54 (2) (2005) 165–175. [12] S. Latifi, M. Hegde, M. Naraghi-Pour, Conditional connectivity measures for large multiprocessor systems, IEEE Trans. Comput. 43 (2) (1994) 218–222. [13] X.Y. Li, J.X. Fan, C.K. Lin, X.H. Jia, Diagnosability evaluation of the data center network DCell, Comput. J. 61 (1) (2018) 129–143. [14] D.S. Li, M. Lu, The g-good-neighbor conditional diagnosability of star graphs under the PMC and MM∗ model, Theoret. Comput. Sci. 674 (2017) 53–59. [15] X.J. Li, J.M. Xu, Edge-fault tolerance of hypercube-like networks, Inf. Process. Lett. 113 (2013) 760–763. [16] L. Lin, L. Xu, S. Zhou, Conditional diagnosability and strong diagnosability of split-star networks under the PMC model, Theoret. Comput. Sci. 562 (2015) 565–580. [17] S.L. Peng, C.K. Lin, J.J.M. Tan, L.H. Hsu, The g-good-neighbor conditional diagnosability of hypercube under PMC model, Appl. Math. Comput. 218 (21) (2012) 10406–10412. [18] F.P. Preparata, G. Metze, R.T. Chien, On the connection assignment problem of diagnosable systems, IEEE Trans. Electron. Comput. EC-16 (1967) 848–854. [19] A. Sengupta, A.T. Dahbura, On self-diagnosable multiprocessor systems: diagnosis by the comparison approach, IEEE Trans. Comput. 41 (11) (1992) 1386–1396. [20] A.S. Vaidya, P.S.N. Rao, S.R. Shankar, A class of hypercube-like networks, in: Proceedings of Fifth IEEE Symposium on Parallel and Distributed Processing, 1993, pp. 800–803. [21] M. Wang, Y. Guo, S. Wang, The 1-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM∗ model, Int. J. Comput. Math. 94 (3) (2016) 620–631. [22] S. Wang, W. Han, The g-good-neighbor conditional diagnosability of n-dimensional hypercubes under the MM∗ model, Inf. Process. Lett. 116 (2016) 574–577. [23] M. Wang, Y. Liu, S. Wang, The 2-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM∗ model, Theor. Comput. Sci. 628 (2016) 92–100.
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