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The g-good-neighbor and g-extra diagnosability of networks
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Theoretical Computer Science www.elsevier.com/locate/tcs
The g-good-neighbor and g-extra diagnosability of networks ✩ Shiying Wang a,∗ , Mujiangshan Wang b a b
School of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, PR China School of Electrical Engineering and Computer Science, The University of Newcastle, NSW 2308, Australia
a r t i c l e
i n f o
Article history: Received 14 March 2018 Received in revised form 25 August 2018 Accepted 7 September 2018 Available online xxxx Communicated by L.M. Kirousis Keywords: Interconnection network Connectivity Diagnosability
a b s t r a c t Diagnosability of a multiprocessor system is an important research topic. The system and interconnection network has a underlying topology, which usually presented by a graph G = ( V , E ). In 2012, a measurement for fault tolerance of the graph was proposed by Peng et al. This measurement is called the g-good-neighbor diagnosability that restrains every fault-free node to contain at least g fault-free neighbors. In 2016, Zhang et al. proposed a new measurement for fault diagnosis of the graph, namely, the g-extra diagnosability, which restrains that every fault-free component has at least ( g + 1) fault-free nodes. A fault set F ⊆ V is called a g-good-neighbor faulty set if the degree d( v ) ≥ g for every vertex v in G − F . A g-good-neighbor cut of G is a g-good-neighbor faulty set F such that G − F is disconnected. The minimum cardinality of g-good-neighbor cuts is said to be the g-good-neighbor connectivity of G. A fault set F ⊆ V is called a g-extra faulty set if every component of G − F has at least ( g + 1) vertices. A g-extra cut of G is a g-extra faulty set F such that G − F is disconnected. The minimum cardinality of g-extra cuts is said to be the g-extra connectivity of G. The g-good-neighbor (extra) diagnosability and g-goodneighbor (extra) connectivity of many well-known graphs have been widely investigated. In this paper, we show the relationship between the g-good-neighbor (extra) diagnosability and g-good-neighbor (extra) connectivity of graphs. © 2018 Elsevier B.V. All rights reserved.
1. Introduction A multiprocessor system and interconnection network (networks for short) has a underlying topology, which usually presented by a graph, where nodes represent processors and links represent communication links between processors. Some processors may fail in the system, so processor fault identification plays an important role for reliable computing. The first step to deal with faults is to identify the faulty processors from the fault-free ones. The identification process is called the diagnosis of the system. A system G is said to be t-diagnosable if all faulty processors can be identified without replacing the faulty processors, provided that the number of faulty processors presented does not exceed t. The diagnosability t (G ) of G is the maximum value of t such that G is t-diagnosable [2,3,5]. For a t-diagnosable system, Dahbura and Masson [2] proposed an algorithm with time complex O (n2.5 ), which can effectively identify the set of faulty processors. Several diagnosis models were proposed to identify the faulty processors. One of most commonly uses is the Preparata, Metze, and Chien’s (PMC) diagnosis model introduced by Preparata et al. [8]. The diagnosis of the system is achieved through two linked processors testing each other. Similar issue, namely the comparison diagnosis model (MM model), was proposed
✩ This work is supported by the National Natural Science Foundation of China (61772010) and the Science Foundation of Henan Normal University (Xiao 20180529, 20180454). Corresponding author. E-mail addresses: [email protected], [email protected] (S. Wang).
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https://doi.org/10.1016/j.tcs.2018.09.002 0304-3975/© 2018 Elsevier B.V. All rights reserved.
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by Maeng and Malek [6]. In the MM model, to diagnose a system, a node sends the same task to two of its neighbors, and then compares their responses. The MM* is a special case of the MM model and each node must test its any pair of adjacent nodes of the system. In 2005, Lai et al. [5] introduced a measurement for fault diagnosis of a system, namely, the conditional diagnosability. They consider the situation that no fault set can contain all the neighbors of any vertex in the system. In 2012, Peng et al. [7] proposed a measurement for fault diagnosis of the system, namely, the g-good-neighbor diagnosability (which is also called the g-good-neighbor conditional diagnosability), which requires that every fault-free node has at least g fault-free neighbors. In [7], they studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the PMC model. In [13], Wang and Han studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the MM∗ model. In 2016, Xu et al. [17] studied the g-good-neighbor diagnosability of complete cubic networks under the PMC model and MM∗ model. Yuan et al. [18,19] studied that the g-good-neighbor diagnosability of the k-ary n-cube (k ≥ 3) under the PMC model and MM∗ model. The Cayley graph C n generated by the transposition tree n has recently received considerable attention. In 2016 and 2017, Wang et al. [10,11] studied the g-good-neighbor diagnosability of C n under the PMC model and MM∗ model for g ∈ {1, 2}. The n-dimensional bubble-sort star graph B S n has many good properties. In 2017, Wang et al. [15] studied the 2-good-neighbor connectivity and 2-good-neighbor diagnosability of B S n . In 2018, Wang et al. [12] studied the 1-good-neighbor connectivity and diagnosability of Cayley graphs generated by complete graphs. In 2016, Zhang et al. [20] proposed a new measure for fault diagnosis of the system, namely, the g-extra diagnosability, which restrains that every fault-free component has at least ( g + 1) fault-free nodes. In [20], they studied the g-extra diagnosability of the n-dimensional hypercube under the PMC model and MM∗ model. In 2016, Wang et al. [14] studied that the 2-extra diagnosability of B S n under the PMC model and MM∗ model. In 2017, Wang and Yang [16] studied the 2-good-neighbor (2-extra) diagnosability of alternating group graph networks under the PMC model and MM∗ model. In this paper, we show (1) the relationship between the g-extra diagnosability and g-extra connectivity of graphs under the PMC model; (2) the relationship between the g-extra diagnosability and g-extra connectivity of graphs under the MM∗ model; (3) the relationship between the g-good-neighbor diagnosability and g-good-neighbor connectivity of graphs under the PMC model; (4) the relationship between the g-good-neighbor diagnosability and g-good-neighbor connectivity of graphs under the MM∗ model. 2. Preliminaries In this section, some definitions and notations needed for our discussion, the PMC model and MM* model are introduced, and so on. 2.1. Notations A multiprocessor system and network is modeled as an undirected simple graph G = ( V , E ), whose vertices (nodes) represent processors and edges (links) represent communication links. Given a nonempty vertex subset V of V , the induced subgraph by V in G, denoted by G [ V ], is a graph, whose vertex set is V and the edge set is the set of all the edges of G with both endpoints in V . The degree d G ( v ) of a vertex v is the number of edges incident with v. The minimum degree of a vertex in G is denoted by δ(G ). For any vertex v, we define the neighborhood N G ( v ) of v in G to be the set of vertices adjacent to v. For u ∈ N G ( v ), u is called a neighbor vertex or a neighbor of v. Let S ⊆ V . We use N G ( S ) to denote the set ∪ v ∈ S N G ( v )\ S. For neighborhoods and degrees, we will usually omit the subscript for the graph when no confusion arises. A graph G is said to be k-regular if for any vertex v, d G ( v ) = k. Let G be connected. The connectivity κ (G ) of G is the minimum number of vertices whose removal results in a disconnected graph or only one vertex left when G is complete. Let F 1 and F 2 be two distinct subsets of V , and let the symmetric difference F 1 F 2 = ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 ). For graph-theoretical terminology and notation not defined here, we follow [1]. Let G = ( V , E ) be connected. A fault set F ⊆ V is called a g-good-neighbor faulty set if | N ( v ) ∩ ( V \ F )| ≥ g for every vertex v in V \ F . A g-good-neighbor cut of G is a g-good-neighbor faulty set F such that G − F is disconnected. The minimum cardinality of g-good-neighbor cuts is said to be the g-good-neighbor connectivity of G, denoted by κ ( g ) (G ). A connected graph G is said to be g-good-neighbor connected if G has a g-good-neighbor cut. A fault set F ⊆ V is called a g-extra faulty set if every component of G − F has at least ( g + 1) vertices. A g-extra cut of G is a g-extra faulty set F such that G − F is disconnected. The minimum cardinality of g-extra cuts is said to be the g-extra connectivity of G, denoted by κ˜ ( g ) (G ). A connected graph G is said to be g-extra connected if G has a g-extra cut. Proposition 2.1. ([9]) Let G be a g-extra and g-good-neighbor connected graph. Then κ˜ ( g ) (G ) ≤ κ ( g ) (G ). Proposition 2.2. ([9]) Let G be a 1-good-neighbor connected graph. Then κ (1) (G ) = κ˜ (1) (G ). 2.2. The PMC model and M M ∗ model Under the PMC model [6,18], to diagnose a system G = ( V (G ), E (G )), two adjacent nodes in G are capable to perform tests on each other. For two adjacent nodes u and v in V (G ), the test performed by u on v is represented by the ordered
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pair (u , v ). The outcome of a test (u , v ) is 1 (resp. 0) if u evaluate v as faulty (resp. fault-free). We assume that the test result is reliable (resp. unreliable) if the node u is fault-free (resp. faulty). A test assignment T for G is a collection of tests for every adjacent pair of vertices. It can be modeled as a directed testing graph T = ( V (G ), L ), where (u , v ) ∈ L implies that u and v are adjacent in G. The collection of all test results for a test assignment T is called a syndrome. Formally, a syndrome is a function σ : L → {0, 1}. The set of all faulty processors in G is called a faulty set. This can be any subset of V (G ). For a given syndrome σ , a subset of vertices F ⊆ V (G ) is said to be consistent with σ if syndrome σ can be produced from the situation that, for any (u , v ) ∈ L such that u ∈ V \ F , σ (u , v ) = 1 if and only if v ∈ F . This means that F is a possible set of faulty processors. Since a test outcome produced by a faulty processor is unreliable, a given set F of faulty vertices may produce a lot of different syndromes. On the other hand, different faulty sets may produce the same syndrome. Let σ ( F ) denote the set of all syndromes which F is consistent with. Under the PMC model, two distinct sets F 1 and F 2 in V (G ) are said to be indistinguishable if σ ( F 1 ) ∩ σ ( F 2 ) = ∅, otherwise, F 1 and F 2 are said to be distinguishable. Besides, we say ( F 1 , F 2 ) is an indistinguishable pair if σ ( F 1 ) ∩ σ ( F 2 ) = ∅; else, ( F 1 , F 2 ) is a distinguishable pair. Using the MM model, the diagnosis is carried out by sending the same testing task to a pair of processors and comparing their responses. We always assume the output of a comparison performed by a faulty processor is unreliable. The comparison scheme of a system G = ( V (G ), E (G )) is modeled as a multigraph, denoted by M = ( V (G ), L ), where L is the labeled-edge set. A labeled edge (u , v ) w ∈ L represents a comparison in which two vertices u and v are compared by a vertex w, which implies u w , v w ∈ E (G ). The collection of all comparison results in M = ( V (G ), L ) is called the syndrome, denoted by σ ∗ , of the diagnosis. If the comparison (u , v ) w disagrees, then σ ∗ ((u , v ) w ) = 1, otherwise, σ ∗ ((u , v ) w ) = 0. Hence, a syndrome is a function from L to {0, 1}. The MM* model is a special case of the MM model. In the MM* model, all comparisons of G are in the comparison scheme of G, i.e., if u w , v w ∈ E (G ), then (u , v ) w ∈ L. Similarly to the PMC model, we can define a subset of vertices F ⊆ V (G ) is consistent with a given syndrome σ ∗ and two distinct sets F 1 and F 2 in V (G ) are indistinguishable (resp. distinguishable) under the MM* model. A system G = ( V , E ) is g-good-neighbor t-diagnosable if F 1 and F 2 are distinguishable for each distinct pair of g-goodneighbor faulty subsets F 1 and F 2 of V with | F 1 | ≤ t and | F 2 | ≤ t. The g-good-neighbor diagnosability t g (G ) of G is the maximum value of t such that G is g-good-neighbor t-diagnosable. Proposition 2.3. ([7]) For any given system G, t g (G ) ≤ t g (G ) if g ≤ g . In a system G = ( V , E ), a faulty set F ⊆ V is called a conditional faulty set if it does not contain all the neighbor vertices of any vertex in G. A system G is conditional t-diagnosable if for every two distinct conditional faulty subsets F 1 , F 2 ⊆ V with | F 1 | ≤ t , | F 2 | ≤ t, F 1 and F 2 are distinguishable. The conditional diagnosability t c (G ) of G is the maximum number of t such that G is conditional t-diagnosable. By [4], t c (G ) ≥ t (G ). Theorem 2.4. [10] For a system G = ( V , E ), t (G ) = t 0 (G ) ≤ t 1 (G ) ≤ t c (G ). In [10], Wang et al. proved that the 1-good-neighbor diagnosability of the Bubble-sort graph B n under the PMC model is 2n − 3 for n ≥ 4. In [21], Zhou et al. proved the conditional diagnosability of B n is 4n − 11 for n ≥ 4 under the PMC model. Therefore, t 1 ( B n ) < t c ( B n ) when n ≥ 5 and t 1 ( B n ) = t c ( B n ) when n = 4. In a system G = ( V , E ), a faulty set F ⊆ V is called a g-extra faulty set if every component of G − F has more than g nodes. G is g-extra t-diagnosable if and only if for each pair of distinct faulty g-extra vertex subsets F 1 , F 2 ⊆ V (G ) such that | F i | ≤ t, F 1 and F 2 are distinguishable. The g-extra diagnosability of G, denoted by t˜ g (G ), is the maximum value of t such that G is g-extra t-diagnosable. Proposition 2.5. [14] For any given system G, t˜ g (G ) ≤ t˜ g (G ) if g ≤ g . Theorem 2.6. [14] For a system G, t (G ) = t˜0 (G ) ≤ t˜ g (G ) ≤ t g (G ). In particular, t˜1 (G ) = t 1 (G ). 3. The relationship between the g-extra diagnosability and g-extra connectivity of networks under the PMC model and MM∗ model Before discussing the g-extra diagnosability of networks under the PMC model and MM∗ model, we first give two theorems. Theorem 3.1. ([18]) A system G = ( V , E ) is g-extra (g-good-neighbor) t-diagnosable under the P MC model if and only if there is an edge uv ∈ E with u ∈ V \( F 1 ∪ F 2 ) and v ∈ F 1 F 2 for each distinct pair of g-extra (g-good-neighbor) faulty subsets F 1 and F 2 of V with | F 1 | ≤ t and | F 2 | ≤ t (see Fig. 1). Theorem 3.2. ([18]) A system G = ( V , E ) is g-extra (g-good-neighbor) t-diagnosable under the M M ∗ model if and only if each distinct pair of g-extra (g-good-neighbor) faulty subsets F 1 and F 2 of V with | F 1 | ≤ t and | F 2 | ≤ t satisfies one of the following conditions.
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Fig. 1. Illustration of a distinguishable pair ( F 1 , F 2 ) under the P MC model.
Fig. 2. Illustration of a distinguishable pair ( F 1 , F 2 ) under the M M ∗ model.
(1) There are two vertices u , w ∈ V \ ( F 1 ∪ F 2 ) and there is a vertex v ∈ F 1 F 2 such that u w ∈ E and v w ∈ E. (2) There are two vertices u , v ∈ F 1 \ F 2 and there is a vertex w ∈ V \ ( F 1 ∪ F 2 ) such that u w ∈ E and v w ∈ E. (3) There are two vertices u , v ∈ F 2 \ F 1 and there is a vertex w ∈ V \ ( F 1 ∪ F 2 ) such that u w ∈ E and v w ∈ E (see Fig. 2). Theorem 3.3. Let G = ( V (G ), E (G )) be a g-extra connected graph. If there is connected subgraph H of G with | V (G )| = g + 1 such that N ( V ( H )) is a minimum g-extra cut of G, then the g-extra diagnosability of G is less than or equal to κ˜ ( g ) (G ) + g, i.e., t˜ g (G ) ≤ κ˜ ( g ) (G ) + g under the PMC model and MM∗ model. Proof. Since N ( V ( H )) is a minimum g-extra cut of G, | N ( V ( H ))| = κ˜ ( g ) (G ) holds. Let F 1 = N ( V ( H )), and let F 2 = F 1 + V ( H ). Then | F 2 | = κ˜ ( g ) (G ) + g + 1. Therefore, F 1 and F 2 are both g-extra faulty sets of G with | F 1 | = κ˜ ( g ) (G ) and | F 2 | = κ˜ ( g ) (G ) + g + 1. Since V ( H ) = F 1 F 2 and F 1 ⊂ F 2 , there is no edge of G between V (G )\( F 1 ∪ F 2 ) and F 1 F 2 . By Theorems 3.1 and 3.2, we can deduce that G is not g-extra (κ˜ ( g ) (G ) + g + 1)-diagnosable under the PMC model and MM∗ model. Hence, by the definition of g-extra diagnosability, we conclude that the g-extra diagnosability of G is less than to κ˜ ( g ) (G ) + g + 1, i.e., t˜ g (G ) ≤ κ˜ ( g ) (G ) + g. 2 Theorem 3.4. Let G = ( V (G ), E (G )) be a g-extra connected graph, and let V (G ) = F 1 ∪ F 2 for each distinct pair of g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g and | F 2 | ≤ κ˜ ( g ) (G ) + g. Then the g-extra diagnosability of G is more than or equal to κ˜ ( g ) (G ) + g, i.e., t g (G ) ≥ κ˜ ( g ) (G ) + g under the PMC model. Proof. By the definition of g-extra diagnosability, it is sufficient to show that G is g-extra (κ˜ ( g ) (G ) + g )-diagnosable. By Theorem 3.1, suppose, on the contrary, that there are two distinct g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g and | F 2 | ≤ κ˜ ( g ) (G ) + g, but the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition in Theorem 3.1, i.e., there are no edges between V (G )\( F 1 ∪ F 2 ) and F 1 F 2 . Without loss of generality, assume that F 2 \ F 1 = ∅. Since there are no edges between V (G ) \ ( F 1 ∪ F 2 ) and F 1 F 2 , and F 1 is a g-extra faulty set, G − F 1 has two parts G − F 1 − F 2 and G [ F 2 \ F 1 ] (for convenience). Thus, every component G i of G − F 1 − F 2 has | V (G i )| ≥ g + 1 and every component B i of G [ F 2 \ F 1 ]) has | V ( B i )| ≥ g + 1. Similarly, every component B i of G [ F 1 \ F 2 ]) has | V ( B )| ≥ g + 1 when F 1 \ F 2 = ∅. Therefore, F 1 ∩ F 2 is also a g-extra faulty set of G. Note that F 1 ∩ F 2 = F 1 is also a g-extra faulty set when F 1 \ F 2 = ∅. Since there are no edges between V (G − F 1 − F 2 ) and F 1 F 2 , F 1 ∩ F 2 is a g-extra cut of G. If F 1 ∩ F 2 = ∅, this is a contradiction to that G is connected. Therefore, F 1 ∩ F 2 = ∅. Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ g + 1 + κ˜ ( g ) (G ), which contradicts with that | F 2 | ≤ κ˜ ( g ) (G ) + g. So G is g-extra (κ˜ ( g ) (G ) + g )-diagnosable. By the definition of t˜ g (G ), t˜ g (G ) ≥ κ˜ ( g ) (G ) + g. 2 By Theorems 3.3 and 3.4, we have the following theorem. Theorem 3.5. Let G = ( V (G ), E (G )) be a g-extra connected graph, and let V (G ) = F 1 ∪ F 2 for each distinct pair of g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g and | F 2 | ≤ κ˜ ( g ) (G ) + g. If there is connected subgraph H of G with | V ( H )| = g + 1 such that N ( V ( H )) is a minimum g-extra cut of G, then the g-extra diagnosability of G is κ˜ ( g ) (G ) + g under the PMC model. Lemma 3.6. ([14]) Let B S n be the bubble-sort star graph and A = {(1), (12), (123)}. If n ≥ 5, F 1 = N ( A ), F 2 = A ∪ N ( A ), then | F 1 | = 6n − 15, | F 2 | = 6n − 12, F 1 is a 2-extra cut of B S n , and B S n − F 1 has two components B S n − F 2 and B S n [ A ]. Theorem 3.7. ([14]) For n ≥ 5, the 2-extra connectivity of the bubble-sort star graph B S n is 6n − 15.
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By Lemma 3.6, there is connected subgraph B S n [ A ] of order 3 such that N ( A ) is a minimum 2-extra cut of B S n . By Theorem 3.7, κ˜ (2) ( B S n ) = 6n − 15. Since n! > [(6n − 15) + 2] + [(6n − 15) + 2] when n ≥ 5, we have V ( B S n ) = F 1 ∪ F 2 for each distinct pair of 2-extra faulty subsets F 1 and F 2 of B S n with | F 1 | ≤ 6n − 15 + 2 and | F 2 | ≤ 6n − 15 + 2. By Theorem 3.5, we have the following corollary. Corollary 3.8. ([14]) For n ≥ 5, the 2-extra diagnosability of the bubble-sort star graph B S n is 6n − 13 under the PMC model. Let G = ( V (G ), E (G )) be a g-extra connected graph. Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ) for each distinct pair of g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g − 1 and | F 2 | ≤ κ˜ ( g ) (G ) + g − 1. Theorem 3.9. Let G be a g-extra connected graph, and let V ( H ) = ∅ for each distinct pair of g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g − 1 and | F 2 | ≤ κ˜ ( g ) (G ) + g − 1. Then the g-extra diagnosability of G is more than or equal to κ˜ ( g ) (G ) + g − 1, i.e., t g (G ) ≥ κ˜ ( g ) (G ) + g − 1 under the MM∗ model. Proof. By the definition of g-extra diagnosability, it is sufficient to show that G is g-extra (κ˜ ( g ) (G ) + g − 1)-diagnosable. Suppose, on the contrary, that there are two distinct g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g − 1 and | F 2 | ≤ κ˜ ( g ) (G ) + g − 1, but the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 3.2. Without loss of generality, suppose that F 2 \ F 1 = ∅. Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ). Then V ( H ) = ∅. We consider the following cases. Case 1. g = 0. Note that V ( H ) = ∅ for each distinct pair of 0-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ (0) (G ) − 1 and | F 2 | ≤ κ˜ (0) (G ) − 1 and F 2 \ F 1 = ∅. Since the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition (1) of Theorem 3.2, and any vertex of V ( H ) is not isolated in H , we deduce that there is no edge between V ( H ) and F 1 F 2 . Therefore, F 1 ∩ F 2 is a 0-good-neighbor cut of G. Thus, | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ 1 + κ˜ (0) (G ), which contradicts | F 2 | ≤ κ˜ (0) (G ) − 1. Case 2. g ≥ 1. Claim 1. G − F 1 − F 2 has no isolated vertex. Suppose, on the contrary, that G − F 1 − F 2 has at least one isolated vertex w 1 . Since F 1 is a g-extra faulty set, there is a vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . Meanwhile, since the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 3.2, by the condition (3) of Theorem 3.2, there is at most one vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . Thus, there is just a vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . If F 1 \ F 2 = ∅, then F 1 ⊆ F 2 . Since F 2 is a g-extra faulty set, every component G i of G − F 1 − F 2 = G − F 2 satisfies | V (G i )| ≥ g + 1. Therefore, G − F 1 − F 2 has no isolated vertex for g ≥ 1. Thus, F 1 \ F 2 = ∅. Similarly, we know that there is just a vertex a ∈ F 1 \ F 2 such that a is adjacent to w 1 . Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ). Then V ( H ) = ∅. Since the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition (1) of Theorem 3.2, and none of the vertices in V ( H ) is isolated vertex in H , we know that there is no edge between V ( H ) and F 1 F 2 . Note F 2 \ F 1 = ∅. If F 1 ∩ F 2 = ∅, then this is a contradiction to that G is connected. Therefore, F 1 ∩ F 2 = ∅. Thus, F 1 ∩ F 2 is a vertex cut of G. Since F 1 is a g-extra faulty set of G, we have that every component H i of H satisfies | V ( H i )| ≥ g + 1 and every component B i of G [ W ∪( F 2 \ F 1 )]) satisfies | V ( B i )| ≥ g + 1. Since F 2 is a g-extra faulty set of G, we have that every component B i of G [ W ∪ ( F 1 \ F 2 )]) has | V ( B i )| ≥ g + 1. Note that G − ( F 1 ∩ F 2 ) has two parts (for convenience): H and G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]). Let Bi be a component of G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]) and let b i ∈ V (Bi ). If b i ∈ W , then there is a component G i of G ([ F 2 \ F 1 ]) (| V (G i )| ≥ g + 1) and a component B i of G ([ F 1 \ F 2 ]) (| V ( B i )| ≥ g + 1) such that b i ∈ V (G i ) and b i ∈ V ( B i ). It follows that G i ∪ B i is connected in G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]) and b i ∈ V (G i ∪ B i ). Since a connection is an equivalence relation on the vertex set W ∪( F 1 \ F 2 ) ∪( F 2 \ F 1 ), Bi = (G i ∪ B i ) holds. Therefore, | V (Bi )| ≥ g + 1. If b i ∈ ( F 2 \ F 1 ), then there is a component G i of G ([ F 2 \ F 1 ]) (| V (G i )| ≥ g + 1) such that b i ∈ V (G i ). It follows that G i is connected in G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]) and b i ∈ V (G i ). Since a connection is an equivalence relation on the vertex set W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 ), we have that G i is a subgraph of Bi . Therefore, | V (Bi )| ≥ g + 1. Similarly, if b i ∈ ( F 1 \ F 2 ), then | V (Bi )| ≥ g + 1. Therefore, F 1 ∩ F 2 is a g-extra cut of G. Since every component B i of G [ W ∪ ( F 2 \ F 1 )]) has | V ( B i )| ≥ g + 1, we have | F 2 \ F 1 | ≥ g and we have that κ˜ ( g ) (G ) + g − 1 ≥ | F 2 | = | F 1 ∩ F 2 | + | F 2 \ F 1 | ≥ κ˜ ( g ) (G ) + g, a contradiction. The proof of Claim 1 is complete. Let u ∈ V (G ) \ ( F 1 ∪ F 2 ). By Claim 1, u has at least one neighbor vertex in G − F 1 − F 2 . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with any one condition in Theorem 3.2, by the condition (1) of Theorem 3.2, for any pair of adjacent vertices u , w ∈ V (G ) \ ( F 1 ∪ F 2 ), there is no vertex v ∈ F 1 F 2 such that u w ∈ E (G ) and v w ∈ E (G ). It follows that u has no neighbor in F 1 F 2 . Since u is taken arbitrarily, so there is no edge between V (G ) \ ( F 1 ∪ F 2 ) and F 1 F 2 . If F 1 ∩ F 2 = ∅, then this is a contradiction to the assumption that G is connected. Therefore, F 1 ∩ F 2 = ∅ and F 1 ∩ F 2 is a cut of G. Since
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[m3G; v1.243; Prn:14/09/2018; 14:03] P.6 (1-8)
S. Wang, M. Wang / Theoretical Computer Science ••• (••••) •••–•••
F 2 \ F 1 = ∅ and F 1 is a g-extra faulty set, we have that every component H i of G − F 1 − F 2 has | V ( H i )| ≥ g + 1 and every component G i of G ([ F 2 \ F 1 ]) has | V (G i )| ≥ g + 1. Suppose that F 1 \ F 2 = ∅. Then F 1 ∩ F 2 = F 1 . Since F 1 is a g-extra faulty set of G, we have that F 1 ∩ F 2 = F 1 is a g-extra faulty set of G. Since there is no edge between V (G ) \ ( F 1 ∪ F 2 ) and F 2 \ F 1 , we have that F 1 ∩ F 2 = F 1 is a g-extra cut of G. Suppose that F 1 \ F 2 = ∅. Similarly, every component B i of G ([ F 1 \ F 2 ]) has | V ( B i )| ≥ g + 1. Note that G − ( F 1 ∩ F 2 ) has three parts (for convenience): H , G [ F 1 \ F 2 ] and G [ F 2 \ F 1 ]. Therefore, F 1 ∩ F 2 is a g-extra cut of G. Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ g + 1 + κ˜ ( g ) (G ), which contradicts | F 2 | ≤ κ˜ ( g ) (G ) + g − 1. Therefore, G is g-extra (κ˜ ( g ) (G ) + g − 1)-diagnosable and t˜ g (G ) ≥ κ˜ ( g ) (G ) + g − 1. The proof is complete. 2 By Theorems 3.3 and 3.9, we have the following theorem. Theorem 3.10. Let G be a g-extra connected graph, and let V ( H ) = ∅ for each distinct pair of g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g and | F 2 | ≤ κ˜ ( g ) (G ) + g. If there is connected subgraph H of G with | V ( H )| = g + 1 such that N ( V ( H )) is a minimum g-extra cut of G, then the g-extra diagnosability of G is κ˜ ( g ) (G ) + g − 1 or κ˜ ( g ) (G ) + g under the MM∗ model. 4. The relationship between the g-good-neighbor diagnosability and g-good-neighbor connectivity of networks under the PMC model and MM∗ model In this section, we will give the relationship between the g-good-neighbor diagnosability and g-good-neighbor connectivity of networks under the PMC and MM∗ model. Theorem 4.1. Let G = ( V (G ), E (G )) be a g-good-neighbor connected graph, and let H be connected subgraph of G with δ( H ) = g such that it contains V (G ) as least as possible, and N ( V ( H )) is a minimum g-good-neighbor cut of G. Then the g-good-neighbor diagnosability of G is less than or equal to κ ( g ) (G ) + | V ( H )| − 1, i.e., t g (G ) ≤ κ ( g ) (G ) + | V ( H )| − 1 under the PMC model and MM∗ model. Proof. Since N ( V ( H )) is a minimum g-good-neighbor cut of G, | N ( V ( H ))| = κ ( g ) (G ) holds. Let F 1 = N ( V ( H )), and let F 2 = F 1 + V ( H ). Then | F 2 | = κ ( g ) (G ) + | V ( H )|. Therefore, F 1 and F 2 are both g-good-neighbor faulty sets of G with | F 1 | = κ ( g ) (G ) and | F 2 | = κ ( g ) (G ) + | V ( H )|. Since V ( H ) = F 1 F 2 and F 1 ⊂ F 2 , there is no edge of G between V (G )\( F 1 ∪ F 2 ) and F 1 F 2 . By Theorems 3.1 and 3.2, we know that G is not g-good-neighbor (κ ( g ) (G ) + | V ( H )|)-diagnosable under the PMC model and MM∗ model. Hence, by the definition of g-good-neighbor diagnosability, we conclude that the g-good-neighbor diagnosability of G is less than to κ ( g ) (G ) + | V ( H )|, i.e., t g (G ) ≤ κ ( g ) (G ) + | V ( H )| − 1. 2 Theorem 4.2. Let G = ( V (G ), E (G )) be a g-good-neighbor connected graph, and let H be connected subgraph of G with δ( H ) = g such that it contains V (G ) as least as possible, and V (G ) = F 1 ∪ F 2 for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 1 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 1. Then the g-good-neighbor diagnosability of G is more than or equal to κ ( g ) (G ) + | V ( H )| − 1, i.e., t g (G ) ≥ κ ( g ) (G ) + | V ( H )| − 1 under the PMC model. Proof. By the definition of g-good-neighbor diagnosability, it is sufficient to show that G is g-good-neighbor (κ ( g ) (G ) + | V ( H )| − 1)-diagnosable. By Theorem 3.1, suppose, on the contrary, that there are two distinct g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 1 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 1, but the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition in Theorem 3.1, i.e., there are no edges between V (G )\( F 1 ∪ F 2 ) and F 1 F 2 . Without loss of generality, assume that F 2 \ F 1 = ∅. Since there are no edges between V (G ) \ ( F 1 ∪ F 2 ) and F 1 F 2 , and F 1 is a g-good-neighbor faulty set, G − F 1 has two parts G − F 1 − F 2 and G [ F 2 \ F 1 ] (for convenience). Thus, | N ( v ) ∩ ( V \ ( F 1 ∪ F 2 ))| ≥ g for every vertex v in V \ ( F 1 ∪ F 2 ). By the definition of H , | V (G − F 1 − F 2 )| ≥ | V ( H )| holds. Similarly, | N ( v ) ∩ ( F 2 \ F 1 )| ≥ g for every vertex v in F 2 \ F 1 and | N ( v ) ∩ ( F 1 \ F 2 )| ≥ g for every vertex v in F 1 \ F 2 when F 1 \ F 2 = ∅, and | F 2 \ F 1 | ≥ | V ( H )| and | F 1 \ F 2 | ≥ | V ( H )| when F 1 \ F 2 = ∅. Therefore, F 1 ∩ F 2 is also a g-good-neighbor faulty set of G. Note that F 1 ∩ F 2 = F 1 is also a g-extra faulty set when F 1 \ F 2 = ∅. Since there are no edges between V (G − F 1 − F 2 ) and F 1 F 2 , F 1 ∩ F 2 is a g-good-neighbor cut of G. If F 1 ∩ F 2 = ∅, this is a contradiction to that G is connected. Therefore, F 1 ∩ F 2 = ∅ and hence | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ | V ( H )| + κ ( g ) (G ), which contradicts | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 1. So G is g-good-neighbor (κ ( g ) (G ) + | V ( H )| − 1)-diagnosable. By the definition of t g (G ), t g (G ) ≥ κ ( g ) (G ) + | V ( H )| − 1. 2 By Theorems 4.1 and 4.2, we have the following theorem. Theorem 4.3. Let G = ( V (G ), E (G )) be a g-good-neighbor connected graph, and let H be connected subgraph of G δ(G ) = g such that it contains V (G ) as least as possible and N ( V ( H )) is a minimum g-good-neighbor cut of G, and let H be connected subgraph of G with δ(G ) = g such that it contains V (G ) as least as possible. If V (G ) = F 1 ∪ F 2 for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 1 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 1, then κ ( g ) (G ) + | V ( H )| − 1 ≤ t g (G ) ≤ κ ( g ) (G ) + | V ( H )| − 1 under the PMC model.
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Doctopic: Theory of natural computing
[m3G; v1.243; Prn:14/09/2018; 14:03] P.7 (1-8)
S. Wang, M. Wang / Theoretical Computer Science ••• (••••) •••–•••
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Lemma 4.4. ([15]) Let B S n be the bubble-sort star graph and A = {(1), (12), (123), (13)}. If n ≥ 5, F 1 = N ( A ), F 2 = A ∪ N ( A ), then | F 1 | = 8n − 22, | F 2 | = 8n − 18, δ( B S n − F 1 ) ≥ 2, and δ( B S n − F 2 ) ≥ 2. Theorem 4.5. ([15]) For n ≥ 5, the 2-good-neighbor connectivity of the bubble-sort star graph B S n is 8n − 22. By Lemma 4.4, there is connected subgraph B S n [ A ] of minimum degree 2 such that it contains V ( B S n ) as least as possible and N ( A ) is a minimum 2-good-neighbor cut of B S n By Theorem 4.5, κ (2) ( B S n ) = 8n − 22. Since n! > [(8n − 22) + 4 − 1] + [(8n − 22) + 4 − 1] when n ≥ 5, we have V ( B S n ) = F 1 ∪ F 2 for each distinct pair of 2-good-neighbor faulty subsets F 1 and F 2 of B S n with | F 1 | ≤ (8n − 22) + 4 − 1 and | F 2 | ≤ (8n − 22) + 4 − 1. By Theorem 4.3, we have the following corollary. Corollary 4.6. ([15]) For n ≥ 5, the 2-good-neighbor diagnosability of the bubble-sort star graph B S n is 8n − 19 under the PMC model. Let G = ( V (G ), E (G )) be a g-good-neighbor connected graph. Suppose that H is connected subgraph of G with δ( H ) = g such that it contains V (G ) as least as possible. Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H ∗ be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ) for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 2 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 2. Theorem 4.7. Let G be a g-good-neighbor connected graph, and let V ( H ∗ ) = ∅ for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 2 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 2. Then the g-good-neighbor diagnosability of G is more than or equal to κ ( g ) (G ) + | V ( H )| − 2, i.e., t g (G ) ≥ κ ( g ) (G ) + | V ( H )| − 2 under the MM∗ model. Proof. By the definition of g-good-neighbor diagnosability, it is sufficient to show that G is g-good-neighbor (κ ( g ) (G ) + | V ( H )| − 2)-diagnosable. Suppose, on the contrary, that there are two distinct g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 2 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 2, but the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 3.2. Without loss of generality, suppose that F 2 \ F 1 = ∅. Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H ∗ be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ). Then V ( H ∗ ) = ∅. We consider the following cases. Case 1. g = 0. By the definition of H , | V ( H )| = 1. Note that V ( H ∗ ) = ∅ for each distinct pair of 0-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ (0) (G ) + | V ( H )| − 2 = κ (0) (G ) − 1 and | F 2 | ≤ κ (0) (G ) + | V ( H )| − 2 = κ (0) (G ) − 1 and F 2 \ F 1 = ∅. Since the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition (1) of Theorem 3.2, and any vertex of V ( H ∗ ) is not isolated in H ∗ , we deduce that there is no edge between V ( H ∗ ) and F 1 F 2 . Therefore, F 1 ∩ F 2 is a 0-good-neighbor cut of G. Thus, | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ 1 + κ (0) (G ), which contradicts | F 2 | ≤ κ (0) (G ) − 1. Case 2. g ≥ 1. Claim 1. G − F 1 − F 2 has no isolated vertex. Suppose, on the contrary, that G − F 1 − F 2 has at least one isolated vertex w 1 . Since F 1 is one g-good-neighbor faulty set, there is a vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . Meanwhile, since the vertex set pair ( F 1 , F 2 ) is not satisfied with any one condition in Theorem 3.2, by the condition (3) of Theorem 3.2, there is at most one vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . Thus, there is just a vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . So d( w 1 ) = 1 in G [{ w 1 } ∪ ( F 2 \ F 1 )]. Since F 1 is a g-good-neighbor faulty set, this is a contradiction when g ≥ 2. Then F 1 is a 1-good-neighbor faulty set. If F 1 \ F 2 = ∅, then F 1 ⊆ F 2 . Since F 2 is a g-good-neighbor faulty set, every vertex v of G − F 1 − F 2 = G − F 2 has d( v ) ≥ g in G − F 2 . Therefore, G − F 1 − F 2 has no isolated vertex for g ≥ 1. Thus, F 1 \ F 2 = ∅. Similarly, we know that there is just a vertex a ∈ F 1 \ F 2 such that a is adjacent to w 1 and F 2 is a 1-good-neighbor faulty set. Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H ∗ be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ). Then V ( H ∗ ) = ∅. Since the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition (1) of Theorem 3.2, and any vertex of V ( H ∗ ) is not isolated in H ∗ , we deduce that there is no edge between V ( H ∗ ) and F 1 F 2 . Note F 2 \ F 1 = ∅. If F 1 ∩ F 2 = ∅, then this is a contradiction to that G is connected. Therefore, F 1 ∩ F 2 = ∅. Thus, F 1 ∩ F 2 is a vertex cut of G. Since F 1 is a 1-good-neighbor faulty set of G, we have that every vertex v of H ∗ has d H ∗ ( v ) ≥ 1 and every vertex a of G [ W ∪ ( F 2 \ F 1 )]) has d(a) ≥ 1 in G [ W ∪( F 2 \ F 1 )]). Since F 2 is a 1-good-neighbor faulty set of G, we have that every vertex b of G [ W ∪( F 1 \ F 2 )]) has d(b) ≥ 1 in G [ W ∪ ( F 1 \ F 2 )]). Therefore, every vertex x of G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]) has d(x) ≥ 1 in G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]). Note that G − ( F 1 ∩ F 2 ) has two parts (for convenience): H ∗ and G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]). Therefore, F 1 ∩ F 2 is a 1-good-neighbor cut of G and hence we have that κ (1) (G ) + 2 − 2 ≥ | F 2 | = | F 1 ∩ F 2 | + | F 2 \ F 1 | ≥ κ (1) (G ) + 1, a contradiction. The proof of Claim 1 is complete.
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Doctopic: Theory of natural computing
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S. Wang, M. Wang / Theoretical Computer Science ••• (••••) •••–•••
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Let u ∈ V (G ) \ ( F 1 ∪ F 2 ). By Claim 1, u has at least one neighbor vertex in G − F 1 − F 2 . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 3.2, by the condition (1) of Theorem 3.2, for any pair of adjacent vertices u , w ∈ V (G ) \ ( F 1 ∪ F 2 ), there is no vertex v ∈ F 1 F 2 such that u w ∈ E (G ) and v w ∈ E (G ). It follows that u has no neighbor in F 1 F 2 . Since u is taken arbitrarily, so there is no edge between V (G ) \ ( F 1 ∪ F 2 ) and F 1 F 2 . If F 1 ∩ F 2 = ∅, then this is a contradiction to that G is connected. Therefore, F 1 ∩ F 2 = ∅ and F 1 ∩ F 2 is a cut of G. Since F 2 \ F 1 = ∅ and F 1 is a g-good-neighbor faulty set, we have that every vertex v of G − F 1 − F 2 has d( v ) ≥ g ≥ 1 in G − F 1 − F 2 and every vertex a of G ([ F 2 \ F 1 ]) has d(a) ≥ g ≥ 1 in G ([ F 2 \ F 1 ]). By the definition of H , | F 2 \ F 1 | ≥ | V ( H )|. Suppose that F 1 \ F 2 = ∅. Then F 1 ∩ F 2 = F 1 . Since F 1 is a g-good-neighbor faulty set of G, we have that F 1 ∩ F 2 = F 1 is a g-good-neighbor faulty set of G. Since there is no edge between V (G ) \ ( F 1 ∪ F 2 ) and F 2 \ F 1 , we have that F 1 ∩ F 2 = F 1 is a g-good-neighbor cut of G. Suppose that F 1 \ F 2 = ∅. Similarly, every vertex b of G ([ F 1 \ F 2 ]) has d(b) ≥ g. Note that G − ( F 1 ∩ F 2 ) has three parts (for convenience): H ∗ , G [ F 1 \ F 2 ] and G [ F 2 \ F 1 ]. Therefore, F 1 ∩ F 2 is a g-good-neighbor cut of G and hence | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ | V ( H )| + κ ( g ) (G ), which contradicts | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 2. Therefore, G is g-good-neighbor (κ ( g ) (G ) + | V ( H )| − 2)-diagnosable and t g (G ) ≥ κ ( g ) (G ) + | V ( H )| − 2. The proof is complete. 2 By Theorems 4.1 and 4.7, we have the following theorem. Theorem 4.8. Let G be a g-good-neighbor connected graph, and let H be connected subgraph of G with δ( H ) = g such that it contains V (G ) as least as possible, and N ( V ( H )) is a minimum g-good-neighbor cut of G, and let H be connected subgraph of G with δ(G ) = g such that it contains V (G ) as least as possible. If V ( H ∗ ) = ∅ for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 2 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 2, then κ ( g ) (G ) + | V ( H )| − 2 ≤ t g (G ) ≤ κ ( g ) (G ) + | V ( H )| − 1 under the MM∗ model. 5. Conclusions Conditional connectivity and conditional diagnosability are two important metrics for fault tolerance of a multiprocessor system. In this paper, we show the relationship between the g-good-neighbor (extra) diagnosability and g-good-neighbor (extra) connectivity of networks. It provides a simple way to find the g-good-neighbor (extra) diagnosability of some wellknown networks based on the g-good-neighbor (extra) connectivity. The work will help engineers to develop more different networks. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, New York, 2007. A.T. Dahbura, G.M. Masson, An O (n2.5 ) fault identification algorithm for diagnosable systems, IEEE Trans. Comput. 33 (6) (1984) 486–492. Jianxi Fan, Diagnosability of crossed cubes under the comparison diagnosis model, IEEE Trans. Parallel Distrib. Syst. 13 (10) (2002) 1099–1104. Sun-Yuan Hsieh, Chi-Ya Kao, The conditional diagnosability of k-ary n-cubes under the comparison diagnosis model, IEEE Trans. Comput. 62 (4) (2013) 839–843. Pao-Lien Lai, Jimmy J.M. Tan, Chien-Ping Chang, Lih-Hsing Hsu, Conditional diagnosability measures for large multiprocessor systems, IEEE Trans. Comput. 54 (2) (2005) 165–175. J. Maeng, M. Malek, A comparison connection assignment for self-diagnosis of multiprocessor systems, in: Proceeding of 11th International Symposium on Fault-Tolerant Computing, 1981, pp. 173–175. Shao-Lun Peng, Cheng-Kuan Lin, Jimmy J.M. Tan, Lih-Hsing Hsu, The g-good-neighbor conditional diagnosability of hypercube under PMC model, Appl. Math. Comput. 218 (21) (2012) 10406–10412. F.P. Preparata, G. Metze, R.T. Chien, On the connection assignment problem of diagnosable systems, IEEE Trans. Comput. EC-16 (1967) 848–854. Yunxia Ren, Shiying Wang, Some properties of the g-good-neighbor (g-extra) diagnosability of a multiprocessor system, Am. J. Comput. Math. 6 (2016) 259–266. Mujiangshan Wang, Yubao Guo, Shiying Wang, The 1-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM∗ model, Int. J. Math. Comput. Sci. 94 (3) (2017) 620–631. Mujiangshan Wang, Yuqing Lin, Shiying Wang, The 2-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM* model, Theoret. Comput. Sci. 628 (2016) 92–100. Mujiangshan Wang, Yuqing Lin, Shiying Wang, The 1-good-neighbor connectivity and diagnosability of Cayley graphs generated by complete graphs, Discrete Appl. Math. 246 (2018) 108–118. Shiying Wang, Weiping Han, The g-good-neighbor conditional diagnosability of n-dimensional hypercubes under the MM* model, Inform. Process. Lett. 116 (2016) 574–577. Shiying Wang, Zhenhua Wang, Mujiangshan Wang, The 2-extra connectivity and 2-extra diagnosability of bubble-sort star graph networks, Comput. J. 59 (12) (2016) 1839–1856. Shiying Wang, Zhenhua Wang, Mujiangshan Wang, The 2-good-neighbor connectivity and 2-good-neighbor diagnosability of bubble-sort star graph networks, Discrete Appl. Math. 217 (2017) 691–706. Shiying Wang, Yuxing Yang, The 2-good-neighbor (2-extra) diagnosability of alternating group graph networks under the PMC model and MM* model, Appl. Math. Comput. 305 (2017) 241–250. Xiang Xu, Shuming Zhou, Jinqiang Li, Reliability of complete cubic networks under the condition of g-good-neighbor, Comput. J. 60 (5) (2017) 625–635. Jun Yuan, Aixia Liu, Xue Ma, Xiuli Liu, Xiao Qin, Jifu Zhang, The g-good-neighbor conditional diagnosability of k-ary n-cubes under the PMC model and MM∗ model, IEEE Trans. Parallel Distrib. Syst. 26 (2015) 1165–1177. Jun Yuan, Aixia Liu, Xiao Qin, Jifu Zhang, Jing Li, g-Good-neighbor conditional diagnosability measures for 3-ary n-cube networks, Theoret. Comput. Sci. 622 (2016) 144–162. Shurong Zhang, Weihua Yang, The g-extra conditional diagnosability and sequential t /k-diagnosability of hypercubes, Int. J. Math. Comput. Sci. 93 (3) (2016) 482–497. Shuming Zhou, Jian Wang, Xirong Xu, Jun-Ming Xu, Conditional fault diagnosis of bubble sort graphs under the PMC model, in: Intelligence Computation and Evolutionary Computation, in: AISC, vol. 180, 2013, pp. 53–59.
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Theoretical Computer Science www.elsevier.com/locate/tcs
The g-good-neighbor and g-extra diagnosability of networks ✩ Shiying Wang a,∗ , Mujiangshan Wang b a b
School of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, PR China School of Electrical Engineering and Computer Science, The University of Newcastle, NSW 2308, Australia
a r t i c l e
i n f o
Article history: Received 14 March 2018 Received in revised form 25 August 2018 Accepted 7 September 2018 Available online xxxx Communicated by L.M. Kirousis Keywords: Interconnection network Connectivity Diagnosability
a b s t r a c t Diagnosability of a multiprocessor system is an important research topic. The system and interconnection network has a underlying topology, which usually presented by a graph G = ( V , E ). In 2012, a measurement for fault tolerance of the graph was proposed by Peng et al. This measurement is called the g-good-neighbor diagnosability that restrains every fault-free node to contain at least g fault-free neighbors. In 2016, Zhang et al. proposed a new measurement for fault diagnosis of the graph, namely, the g-extra diagnosability, which restrains that every fault-free component has at least ( g + 1) fault-free nodes. A fault set F ⊆ V is called a g-good-neighbor faulty set if the degree d( v ) ≥ g for every vertex v in G − F . A g-good-neighbor cut of G is a g-good-neighbor faulty set F such that G − F is disconnected. The minimum cardinality of g-good-neighbor cuts is said to be the g-good-neighbor connectivity of G. A fault set F ⊆ V is called a g-extra faulty set if every component of G − F has at least ( g + 1) vertices. A g-extra cut of G is a g-extra faulty set F such that G − F is disconnected. The minimum cardinality of g-extra cuts is said to be the g-extra connectivity of G. The g-good-neighbor (extra) diagnosability and g-goodneighbor (extra) connectivity of many well-known graphs have been widely investigated. In this paper, we show the relationship between the g-good-neighbor (extra) diagnosability and g-good-neighbor (extra) connectivity of graphs. © 2018 Elsevier B.V. All rights reserved.
1. Introduction A multiprocessor system and interconnection network (networks for short) has a underlying topology, which usually presented by a graph, where nodes represent processors and links represent communication links between processors. Some processors may fail in the system, so processor fault identification plays an important role for reliable computing. The first step to deal with faults is to identify the faulty processors from the fault-free ones. The identification process is called the diagnosis of the system. A system G is said to be t-diagnosable if all faulty processors can be identified without replacing the faulty processors, provided that the number of faulty processors presented does not exceed t. The diagnosability t (G ) of G is the maximum value of t such that G is t-diagnosable [2,3,5]. For a t-diagnosable system, Dahbura and Masson [2] proposed an algorithm with time complex O (n2.5 ), which can effectively identify the set of faulty processors. Several diagnosis models were proposed to identify the faulty processors. One of most commonly uses is the Preparata, Metze, and Chien’s (PMC) diagnosis model introduced by Preparata et al. [8]. The diagnosis of the system is achieved through two linked processors testing each other. Similar issue, namely the comparison diagnosis model (MM model), was proposed
✩ This work is supported by the National Natural Science Foundation of China (61772010) and the Science Foundation of Henan Normal University (Xiao 20180529, 20180454). Corresponding author. E-mail addresses: [email protected], [email protected] (S. Wang).
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https://doi.org/10.1016/j.tcs.2018.09.002 0304-3975/© 2018 Elsevier B.V. All rights reserved.
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by Maeng and Malek [6]. In the MM model, to diagnose a system, a node sends the same task to two of its neighbors, and then compares their responses. The MM* is a special case of the MM model and each node must test its any pair of adjacent nodes of the system. In 2005, Lai et al. [5] introduced a measurement for fault diagnosis of a system, namely, the conditional diagnosability. They consider the situation that no fault set can contain all the neighbors of any vertex in the system. In 2012, Peng et al. [7] proposed a measurement for fault diagnosis of the system, namely, the g-good-neighbor diagnosability (which is also called the g-good-neighbor conditional diagnosability), which requires that every fault-free node has at least g fault-free neighbors. In [7], they studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the PMC model. In [13], Wang and Han studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the MM∗ model. In 2016, Xu et al. [17] studied the g-good-neighbor diagnosability of complete cubic networks under the PMC model and MM∗ model. Yuan et al. [18,19] studied that the g-good-neighbor diagnosability of the k-ary n-cube (k ≥ 3) under the PMC model and MM∗ model. The Cayley graph C n generated by the transposition tree n has recently received considerable attention. In 2016 and 2017, Wang et al. [10,11] studied the g-good-neighbor diagnosability of C n under the PMC model and MM∗ model for g ∈ {1, 2}. The n-dimensional bubble-sort star graph B S n has many good properties. In 2017, Wang et al. [15] studied the 2-good-neighbor connectivity and 2-good-neighbor diagnosability of B S n . In 2018, Wang et al. [12] studied the 1-good-neighbor connectivity and diagnosability of Cayley graphs generated by complete graphs. In 2016, Zhang et al. [20] proposed a new measure for fault diagnosis of the system, namely, the g-extra diagnosability, which restrains that every fault-free component has at least ( g + 1) fault-free nodes. In [20], they studied the g-extra diagnosability of the n-dimensional hypercube under the PMC model and MM∗ model. In 2016, Wang et al. [14] studied that the 2-extra diagnosability of B S n under the PMC model and MM∗ model. In 2017, Wang and Yang [16] studied the 2-good-neighbor (2-extra) diagnosability of alternating group graph networks under the PMC model and MM∗ model. In this paper, we show (1) the relationship between the g-extra diagnosability and g-extra connectivity of graphs under the PMC model; (2) the relationship between the g-extra diagnosability and g-extra connectivity of graphs under the MM∗ model; (3) the relationship between the g-good-neighbor diagnosability and g-good-neighbor connectivity of graphs under the PMC model; (4) the relationship between the g-good-neighbor diagnosability and g-good-neighbor connectivity of graphs under the MM∗ model. 2. Preliminaries In this section, some definitions and notations needed for our discussion, the PMC model and MM* model are introduced, and so on. 2.1. Notations A multiprocessor system and network is modeled as an undirected simple graph G = ( V , E ), whose vertices (nodes) represent processors and edges (links) represent communication links. Given a nonempty vertex subset V of V , the induced subgraph by V in G, denoted by G [ V ], is a graph, whose vertex set is V and the edge set is the set of all the edges of G with both endpoints in V . The degree d G ( v ) of a vertex v is the number of edges incident with v. The minimum degree of a vertex in G is denoted by δ(G ). For any vertex v, we define the neighborhood N G ( v ) of v in G to be the set of vertices adjacent to v. For u ∈ N G ( v ), u is called a neighbor vertex or a neighbor of v. Let S ⊆ V . We use N G ( S ) to denote the set ∪ v ∈ S N G ( v )\ S. For neighborhoods and degrees, we will usually omit the subscript for the graph when no confusion arises. A graph G is said to be k-regular if for any vertex v, d G ( v ) = k. Let G be connected. The connectivity κ (G ) of G is the minimum number of vertices whose removal results in a disconnected graph or only one vertex left when G is complete. Let F 1 and F 2 be two distinct subsets of V , and let the symmetric difference F 1 F 2 = ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 ). For graph-theoretical terminology and notation not defined here, we follow [1]. Let G = ( V , E ) be connected. A fault set F ⊆ V is called a g-good-neighbor faulty set if | N ( v ) ∩ ( V \ F )| ≥ g for every vertex v in V \ F . A g-good-neighbor cut of G is a g-good-neighbor faulty set F such that G − F is disconnected. The minimum cardinality of g-good-neighbor cuts is said to be the g-good-neighbor connectivity of G, denoted by κ ( g ) (G ). A connected graph G is said to be g-good-neighbor connected if G has a g-good-neighbor cut. A fault set F ⊆ V is called a g-extra faulty set if every component of G − F has at least ( g + 1) vertices. A g-extra cut of G is a g-extra faulty set F such that G − F is disconnected. The minimum cardinality of g-extra cuts is said to be the g-extra connectivity of G, denoted by κ˜ ( g ) (G ). A connected graph G is said to be g-extra connected if G has a g-extra cut. Proposition 2.1. ([9]) Let G be a g-extra and g-good-neighbor connected graph. Then κ˜ ( g ) (G ) ≤ κ ( g ) (G ). Proposition 2.2. ([9]) Let G be a 1-good-neighbor connected graph. Then κ (1) (G ) = κ˜ (1) (G ). 2.2. The PMC model and M M ∗ model Under the PMC model [6,18], to diagnose a system G = ( V (G ), E (G )), two adjacent nodes in G are capable to perform tests on each other. For two adjacent nodes u and v in V (G ), the test performed by u on v is represented by the ordered
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pair (u , v ). The outcome of a test (u , v ) is 1 (resp. 0) if u evaluate v as faulty (resp. fault-free). We assume that the test result is reliable (resp. unreliable) if the node u is fault-free (resp. faulty). A test assignment T for G is a collection of tests for every adjacent pair of vertices. It can be modeled as a directed testing graph T = ( V (G ), L ), where (u , v ) ∈ L implies that u and v are adjacent in G. The collection of all test results for a test assignment T is called a syndrome. Formally, a syndrome is a function σ : L → {0, 1}. The set of all faulty processors in G is called a faulty set. This can be any subset of V (G ). For a given syndrome σ , a subset of vertices F ⊆ V (G ) is said to be consistent with σ if syndrome σ can be produced from the situation that, for any (u , v ) ∈ L such that u ∈ V \ F , σ (u , v ) = 1 if and only if v ∈ F . This means that F is a possible set of faulty processors. Since a test outcome produced by a faulty processor is unreliable, a given set F of faulty vertices may produce a lot of different syndromes. On the other hand, different faulty sets may produce the same syndrome. Let σ ( F ) denote the set of all syndromes which F is consistent with. Under the PMC model, two distinct sets F 1 and F 2 in V (G ) are said to be indistinguishable if σ ( F 1 ) ∩ σ ( F 2 ) = ∅, otherwise, F 1 and F 2 are said to be distinguishable. Besides, we say ( F 1 , F 2 ) is an indistinguishable pair if σ ( F 1 ) ∩ σ ( F 2 ) = ∅; else, ( F 1 , F 2 ) is a distinguishable pair. Using the MM model, the diagnosis is carried out by sending the same testing task to a pair of processors and comparing their responses. We always assume the output of a comparison performed by a faulty processor is unreliable. The comparison scheme of a system G = ( V (G ), E (G )) is modeled as a multigraph, denoted by M = ( V (G ), L ), where L is the labeled-edge set. A labeled edge (u , v ) w ∈ L represents a comparison in which two vertices u and v are compared by a vertex w, which implies u w , v w ∈ E (G ). The collection of all comparison results in M = ( V (G ), L ) is called the syndrome, denoted by σ ∗ , of the diagnosis. If the comparison (u , v ) w disagrees, then σ ∗ ((u , v ) w ) = 1, otherwise, σ ∗ ((u , v ) w ) = 0. Hence, a syndrome is a function from L to {0, 1}. The MM* model is a special case of the MM model. In the MM* model, all comparisons of G are in the comparison scheme of G, i.e., if u w , v w ∈ E (G ), then (u , v ) w ∈ L. Similarly to the PMC model, we can define a subset of vertices F ⊆ V (G ) is consistent with a given syndrome σ ∗ and two distinct sets F 1 and F 2 in V (G ) are indistinguishable (resp. distinguishable) under the MM* model. A system G = ( V , E ) is g-good-neighbor t-diagnosable if F 1 and F 2 are distinguishable for each distinct pair of g-goodneighbor faulty subsets F 1 and F 2 of V with | F 1 | ≤ t and | F 2 | ≤ t. The g-good-neighbor diagnosability t g (G ) of G is the maximum value of t such that G is g-good-neighbor t-diagnosable. Proposition 2.3. ([7]) For any given system G, t g (G ) ≤ t g (G ) if g ≤ g . In a system G = ( V , E ), a faulty set F ⊆ V is called a conditional faulty set if it does not contain all the neighbor vertices of any vertex in G. A system G is conditional t-diagnosable if for every two distinct conditional faulty subsets F 1 , F 2 ⊆ V with | F 1 | ≤ t , | F 2 | ≤ t, F 1 and F 2 are distinguishable. The conditional diagnosability t c (G ) of G is the maximum number of t such that G is conditional t-diagnosable. By [4], t c (G ) ≥ t (G ). Theorem 2.4. [10] For a system G = ( V , E ), t (G ) = t 0 (G ) ≤ t 1 (G ) ≤ t c (G ). In [10], Wang et al. proved that the 1-good-neighbor diagnosability of the Bubble-sort graph B n under the PMC model is 2n − 3 for n ≥ 4. In [21], Zhou et al. proved the conditional diagnosability of B n is 4n − 11 for n ≥ 4 under the PMC model. Therefore, t 1 ( B n ) < t c ( B n ) when n ≥ 5 and t 1 ( B n ) = t c ( B n ) when n = 4. In a system G = ( V , E ), a faulty set F ⊆ V is called a g-extra faulty set if every component of G − F has more than g nodes. G is g-extra t-diagnosable if and only if for each pair of distinct faulty g-extra vertex subsets F 1 , F 2 ⊆ V (G ) such that | F i | ≤ t, F 1 and F 2 are distinguishable. The g-extra diagnosability of G, denoted by t˜ g (G ), is the maximum value of t such that G is g-extra t-diagnosable. Proposition 2.5. [14] For any given system G, t˜ g (G ) ≤ t˜ g (G ) if g ≤ g . Theorem 2.6. [14] For a system G, t (G ) = t˜0 (G ) ≤ t˜ g (G ) ≤ t g (G ). In particular, t˜1 (G ) = t 1 (G ). 3. The relationship between the g-extra diagnosability and g-extra connectivity of networks under the PMC model and MM∗ model Before discussing the g-extra diagnosability of networks under the PMC model and MM∗ model, we first give two theorems. Theorem 3.1. ([18]) A system G = ( V , E ) is g-extra (g-good-neighbor) t-diagnosable under the P MC model if and only if there is an edge uv ∈ E with u ∈ V \( F 1 ∪ F 2 ) and v ∈ F 1 F 2 for each distinct pair of g-extra (g-good-neighbor) faulty subsets F 1 and F 2 of V with | F 1 | ≤ t and | F 2 | ≤ t (see Fig. 1). Theorem 3.2. ([18]) A system G = ( V , E ) is g-extra (g-good-neighbor) t-diagnosable under the M M ∗ model if and only if each distinct pair of g-extra (g-good-neighbor) faulty subsets F 1 and F 2 of V with | F 1 | ≤ t and | F 2 | ≤ t satisfies one of the following conditions.
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Fig. 1. Illustration of a distinguishable pair ( F 1 , F 2 ) under the P MC model.
Fig. 2. Illustration of a distinguishable pair ( F 1 , F 2 ) under the M M ∗ model.
(1) There are two vertices u , w ∈ V \ ( F 1 ∪ F 2 ) and there is a vertex v ∈ F 1 F 2 such that u w ∈ E and v w ∈ E. (2) There are two vertices u , v ∈ F 1 \ F 2 and there is a vertex w ∈ V \ ( F 1 ∪ F 2 ) such that u w ∈ E and v w ∈ E. (3) There are two vertices u , v ∈ F 2 \ F 1 and there is a vertex w ∈ V \ ( F 1 ∪ F 2 ) such that u w ∈ E and v w ∈ E (see Fig. 2). Theorem 3.3. Let G = ( V (G ), E (G )) be a g-extra connected graph. If there is connected subgraph H of G with | V (G )| = g + 1 such that N ( V ( H )) is a minimum g-extra cut of G, then the g-extra diagnosability of G is less than or equal to κ˜ ( g ) (G ) + g, i.e., t˜ g (G ) ≤ κ˜ ( g ) (G ) + g under the PMC model and MM∗ model. Proof. Since N ( V ( H )) is a minimum g-extra cut of G, | N ( V ( H ))| = κ˜ ( g ) (G ) holds. Let F 1 = N ( V ( H )), and let F 2 = F 1 + V ( H ). Then | F 2 | = κ˜ ( g ) (G ) + g + 1. Therefore, F 1 and F 2 are both g-extra faulty sets of G with | F 1 | = κ˜ ( g ) (G ) and | F 2 | = κ˜ ( g ) (G ) + g + 1. Since V ( H ) = F 1 F 2 and F 1 ⊂ F 2 , there is no edge of G between V (G )\( F 1 ∪ F 2 ) and F 1 F 2 . By Theorems 3.1 and 3.2, we can deduce that G is not g-extra (κ˜ ( g ) (G ) + g + 1)-diagnosable under the PMC model and MM∗ model. Hence, by the definition of g-extra diagnosability, we conclude that the g-extra diagnosability of G is less than to κ˜ ( g ) (G ) + g + 1, i.e., t˜ g (G ) ≤ κ˜ ( g ) (G ) + g. 2 Theorem 3.4. Let G = ( V (G ), E (G )) be a g-extra connected graph, and let V (G ) = F 1 ∪ F 2 for each distinct pair of g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g and | F 2 | ≤ κ˜ ( g ) (G ) + g. Then the g-extra diagnosability of G is more than or equal to κ˜ ( g ) (G ) + g, i.e., t g (G ) ≥ κ˜ ( g ) (G ) + g under the PMC model. Proof. By the definition of g-extra diagnosability, it is sufficient to show that G is g-extra (κ˜ ( g ) (G ) + g )-diagnosable. By Theorem 3.1, suppose, on the contrary, that there are two distinct g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g and | F 2 | ≤ κ˜ ( g ) (G ) + g, but the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition in Theorem 3.1, i.e., there are no edges between V (G )\( F 1 ∪ F 2 ) and F 1 F 2 . Without loss of generality, assume that F 2 \ F 1 = ∅. Since there are no edges between V (G ) \ ( F 1 ∪ F 2 ) and F 1 F 2 , and F 1 is a g-extra faulty set, G − F 1 has two parts G − F 1 − F 2 and G [ F 2 \ F 1 ] (for convenience). Thus, every component G i of G − F 1 − F 2 has | V (G i )| ≥ g + 1 and every component B i of G [ F 2 \ F 1 ]) has | V ( B i )| ≥ g + 1. Similarly, every component B i of G [ F 1 \ F 2 ]) has | V ( B )| ≥ g + 1 when F 1 \ F 2 = ∅. Therefore, F 1 ∩ F 2 is also a g-extra faulty set of G. Note that F 1 ∩ F 2 = F 1 is also a g-extra faulty set when F 1 \ F 2 = ∅. Since there are no edges between V (G − F 1 − F 2 ) and F 1 F 2 , F 1 ∩ F 2 is a g-extra cut of G. If F 1 ∩ F 2 = ∅, this is a contradiction to that G is connected. Therefore, F 1 ∩ F 2 = ∅. Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ g + 1 + κ˜ ( g ) (G ), which contradicts with that | F 2 | ≤ κ˜ ( g ) (G ) + g. So G is g-extra (κ˜ ( g ) (G ) + g )-diagnosable. By the definition of t˜ g (G ), t˜ g (G ) ≥ κ˜ ( g ) (G ) + g. 2 By Theorems 3.3 and 3.4, we have the following theorem. Theorem 3.5. Let G = ( V (G ), E (G )) be a g-extra connected graph, and let V (G ) = F 1 ∪ F 2 for each distinct pair of g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g and | F 2 | ≤ κ˜ ( g ) (G ) + g. If there is connected subgraph H of G with | V ( H )| = g + 1 such that N ( V ( H )) is a minimum g-extra cut of G, then the g-extra diagnosability of G is κ˜ ( g ) (G ) + g under the PMC model. Lemma 3.6. ([14]) Let B S n be the bubble-sort star graph and A = {(1), (12), (123)}. If n ≥ 5, F 1 = N ( A ), F 2 = A ∪ N ( A ), then | F 1 | = 6n − 15, | F 2 | = 6n − 12, F 1 is a 2-extra cut of B S n , and B S n − F 1 has two components B S n − F 2 and B S n [ A ]. Theorem 3.7. ([14]) For n ≥ 5, the 2-extra connectivity of the bubble-sort star graph B S n is 6n − 15.
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By Lemma 3.6, there is connected subgraph B S n [ A ] of order 3 such that N ( A ) is a minimum 2-extra cut of B S n . By Theorem 3.7, κ˜ (2) ( B S n ) = 6n − 15. Since n! > [(6n − 15) + 2] + [(6n − 15) + 2] when n ≥ 5, we have V ( B S n ) = F 1 ∪ F 2 for each distinct pair of 2-extra faulty subsets F 1 and F 2 of B S n with | F 1 | ≤ 6n − 15 + 2 and | F 2 | ≤ 6n − 15 + 2. By Theorem 3.5, we have the following corollary. Corollary 3.8. ([14]) For n ≥ 5, the 2-extra diagnosability of the bubble-sort star graph B S n is 6n − 13 under the PMC model. Let G = ( V (G ), E (G )) be a g-extra connected graph. Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ) for each distinct pair of g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g − 1 and | F 2 | ≤ κ˜ ( g ) (G ) + g − 1. Theorem 3.9. Let G be a g-extra connected graph, and let V ( H ) = ∅ for each distinct pair of g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g − 1 and | F 2 | ≤ κ˜ ( g ) (G ) + g − 1. Then the g-extra diagnosability of G is more than or equal to κ˜ ( g ) (G ) + g − 1, i.e., t g (G ) ≥ κ˜ ( g ) (G ) + g − 1 under the MM∗ model. Proof. By the definition of g-extra diagnosability, it is sufficient to show that G is g-extra (κ˜ ( g ) (G ) + g − 1)-diagnosable. Suppose, on the contrary, that there are two distinct g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g − 1 and | F 2 | ≤ κ˜ ( g ) (G ) + g − 1, but the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 3.2. Without loss of generality, suppose that F 2 \ F 1 = ∅. Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ). Then V ( H ) = ∅. We consider the following cases. Case 1. g = 0. Note that V ( H ) = ∅ for each distinct pair of 0-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ (0) (G ) − 1 and | F 2 | ≤ κ˜ (0) (G ) − 1 and F 2 \ F 1 = ∅. Since the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition (1) of Theorem 3.2, and any vertex of V ( H ) is not isolated in H , we deduce that there is no edge between V ( H ) and F 1 F 2 . Therefore, F 1 ∩ F 2 is a 0-good-neighbor cut of G. Thus, | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ 1 + κ˜ (0) (G ), which contradicts | F 2 | ≤ κ˜ (0) (G ) − 1. Case 2. g ≥ 1. Claim 1. G − F 1 − F 2 has no isolated vertex. Suppose, on the contrary, that G − F 1 − F 2 has at least one isolated vertex w 1 . Since F 1 is a g-extra faulty set, there is a vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . Meanwhile, since the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 3.2, by the condition (3) of Theorem 3.2, there is at most one vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . Thus, there is just a vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . If F 1 \ F 2 = ∅, then F 1 ⊆ F 2 . Since F 2 is a g-extra faulty set, every component G i of G − F 1 − F 2 = G − F 2 satisfies | V (G i )| ≥ g + 1. Therefore, G − F 1 − F 2 has no isolated vertex for g ≥ 1. Thus, F 1 \ F 2 = ∅. Similarly, we know that there is just a vertex a ∈ F 1 \ F 2 such that a is adjacent to w 1 . Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ). Then V ( H ) = ∅. Since the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition (1) of Theorem 3.2, and none of the vertices in V ( H ) is isolated vertex in H , we know that there is no edge between V ( H ) and F 1 F 2 . Note F 2 \ F 1 = ∅. If F 1 ∩ F 2 = ∅, then this is a contradiction to that G is connected. Therefore, F 1 ∩ F 2 = ∅. Thus, F 1 ∩ F 2 is a vertex cut of G. Since F 1 is a g-extra faulty set of G, we have that every component H i of H satisfies | V ( H i )| ≥ g + 1 and every component B i of G [ W ∪( F 2 \ F 1 )]) satisfies | V ( B i )| ≥ g + 1. Since F 2 is a g-extra faulty set of G, we have that every component B i of G [ W ∪ ( F 1 \ F 2 )]) has | V ( B i )| ≥ g + 1. Note that G − ( F 1 ∩ F 2 ) has two parts (for convenience): H and G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]). Let Bi be a component of G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]) and let b i ∈ V (Bi ). If b i ∈ W , then there is a component G i of G ([ F 2 \ F 1 ]) (| V (G i )| ≥ g + 1) and a component B i of G ([ F 1 \ F 2 ]) (| V ( B i )| ≥ g + 1) such that b i ∈ V (G i ) and b i ∈ V ( B i ). It follows that G i ∪ B i is connected in G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]) and b i ∈ V (G i ∪ B i ). Since a connection is an equivalence relation on the vertex set W ∪( F 1 \ F 2 ) ∪( F 2 \ F 1 ), Bi = (G i ∪ B i ) holds. Therefore, | V (Bi )| ≥ g + 1. If b i ∈ ( F 2 \ F 1 ), then there is a component G i of G ([ F 2 \ F 1 ]) (| V (G i )| ≥ g + 1) such that b i ∈ V (G i ). It follows that G i is connected in G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]) and b i ∈ V (G i ). Since a connection is an equivalence relation on the vertex set W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 ), we have that G i is a subgraph of Bi . Therefore, | V (Bi )| ≥ g + 1. Similarly, if b i ∈ ( F 1 \ F 2 ), then | V (Bi )| ≥ g + 1. Therefore, F 1 ∩ F 2 is a g-extra cut of G. Since every component B i of G [ W ∪ ( F 2 \ F 1 )]) has | V ( B i )| ≥ g + 1, we have | F 2 \ F 1 | ≥ g and we have that κ˜ ( g ) (G ) + g − 1 ≥ | F 2 | = | F 1 ∩ F 2 | + | F 2 \ F 1 | ≥ κ˜ ( g ) (G ) + g, a contradiction. The proof of Claim 1 is complete. Let u ∈ V (G ) \ ( F 1 ∪ F 2 ). By Claim 1, u has at least one neighbor vertex in G − F 1 − F 2 . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with any one condition in Theorem 3.2, by the condition (1) of Theorem 3.2, for any pair of adjacent vertices u , w ∈ V (G ) \ ( F 1 ∪ F 2 ), there is no vertex v ∈ F 1 F 2 such that u w ∈ E (G ) and v w ∈ E (G ). It follows that u has no neighbor in F 1 F 2 . Since u is taken arbitrarily, so there is no edge between V (G ) \ ( F 1 ∪ F 2 ) and F 1 F 2 . If F 1 ∩ F 2 = ∅, then this is a contradiction to the assumption that G is connected. Therefore, F 1 ∩ F 2 = ∅ and F 1 ∩ F 2 is a cut of G. Since
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S. Wang, M. Wang / Theoretical Computer Science ••• (••••) •••–•••
F 2 \ F 1 = ∅ and F 1 is a g-extra faulty set, we have that every component H i of G − F 1 − F 2 has | V ( H i )| ≥ g + 1 and every component G i of G ([ F 2 \ F 1 ]) has | V (G i )| ≥ g + 1. Suppose that F 1 \ F 2 = ∅. Then F 1 ∩ F 2 = F 1 . Since F 1 is a g-extra faulty set of G, we have that F 1 ∩ F 2 = F 1 is a g-extra faulty set of G. Since there is no edge between V (G ) \ ( F 1 ∪ F 2 ) and F 2 \ F 1 , we have that F 1 ∩ F 2 = F 1 is a g-extra cut of G. Suppose that F 1 \ F 2 = ∅. Similarly, every component B i of G ([ F 1 \ F 2 ]) has | V ( B i )| ≥ g + 1. Note that G − ( F 1 ∩ F 2 ) has three parts (for convenience): H , G [ F 1 \ F 2 ] and G [ F 2 \ F 1 ]. Therefore, F 1 ∩ F 2 is a g-extra cut of G. Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ g + 1 + κ˜ ( g ) (G ), which contradicts | F 2 | ≤ κ˜ ( g ) (G ) + g − 1. Therefore, G is g-extra (κ˜ ( g ) (G ) + g − 1)-diagnosable and t˜ g (G ) ≥ κ˜ ( g ) (G ) + g − 1. The proof is complete. 2 By Theorems 3.3 and 3.9, we have the following theorem. Theorem 3.10. Let G be a g-extra connected graph, and let V ( H ) = ∅ for each distinct pair of g-extra faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ˜ ( g ) (G ) + g and | F 2 | ≤ κ˜ ( g ) (G ) + g. If there is connected subgraph H of G with | V ( H )| = g + 1 such that N ( V ( H )) is a minimum g-extra cut of G, then the g-extra diagnosability of G is κ˜ ( g ) (G ) + g − 1 or κ˜ ( g ) (G ) + g under the MM∗ model. 4. The relationship between the g-good-neighbor diagnosability and g-good-neighbor connectivity of networks under the PMC model and MM∗ model In this section, we will give the relationship between the g-good-neighbor diagnosability and g-good-neighbor connectivity of networks under the PMC and MM∗ model. Theorem 4.1. Let G = ( V (G ), E (G )) be a g-good-neighbor connected graph, and let H be connected subgraph of G with δ( H ) = g such that it contains V (G ) as least as possible, and N ( V ( H )) is a minimum g-good-neighbor cut of G. Then the g-good-neighbor diagnosability of G is less than or equal to κ ( g ) (G ) + | V ( H )| − 1, i.e., t g (G ) ≤ κ ( g ) (G ) + | V ( H )| − 1 under the PMC model and MM∗ model. Proof. Since N ( V ( H )) is a minimum g-good-neighbor cut of G, | N ( V ( H ))| = κ ( g ) (G ) holds. Let F 1 = N ( V ( H )), and let F 2 = F 1 + V ( H ). Then | F 2 | = κ ( g ) (G ) + | V ( H )|. Therefore, F 1 and F 2 are both g-good-neighbor faulty sets of G with | F 1 | = κ ( g ) (G ) and | F 2 | = κ ( g ) (G ) + | V ( H )|. Since V ( H ) = F 1 F 2 and F 1 ⊂ F 2 , there is no edge of G between V (G )\( F 1 ∪ F 2 ) and F 1 F 2 . By Theorems 3.1 and 3.2, we know that G is not g-good-neighbor (κ ( g ) (G ) + | V ( H )|)-diagnosable under the PMC model and MM∗ model. Hence, by the definition of g-good-neighbor diagnosability, we conclude that the g-good-neighbor diagnosability of G is less than to κ ( g ) (G ) + | V ( H )|, i.e., t g (G ) ≤ κ ( g ) (G ) + | V ( H )| − 1. 2 Theorem 4.2. Let G = ( V (G ), E (G )) be a g-good-neighbor connected graph, and let H be connected subgraph of G with δ( H ) = g such that it contains V (G ) as least as possible, and V (G ) = F 1 ∪ F 2 for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 1 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 1. Then the g-good-neighbor diagnosability of G is more than or equal to κ ( g ) (G ) + | V ( H )| − 1, i.e., t g (G ) ≥ κ ( g ) (G ) + | V ( H )| − 1 under the PMC model. Proof. By the definition of g-good-neighbor diagnosability, it is sufficient to show that G is g-good-neighbor (κ ( g ) (G ) + | V ( H )| − 1)-diagnosable. By Theorem 3.1, suppose, on the contrary, that there are two distinct g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 1 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 1, but the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition in Theorem 3.1, i.e., there are no edges between V (G )\( F 1 ∪ F 2 ) and F 1 F 2 . Without loss of generality, assume that F 2 \ F 1 = ∅. Since there are no edges between V (G ) \ ( F 1 ∪ F 2 ) and F 1 F 2 , and F 1 is a g-good-neighbor faulty set, G − F 1 has two parts G − F 1 − F 2 and G [ F 2 \ F 1 ] (for convenience). Thus, | N ( v ) ∩ ( V \ ( F 1 ∪ F 2 ))| ≥ g for every vertex v in V \ ( F 1 ∪ F 2 ). By the definition of H , | V (G − F 1 − F 2 )| ≥ | V ( H )| holds. Similarly, | N ( v ) ∩ ( F 2 \ F 1 )| ≥ g for every vertex v in F 2 \ F 1 and | N ( v ) ∩ ( F 1 \ F 2 )| ≥ g for every vertex v in F 1 \ F 2 when F 1 \ F 2 = ∅, and | F 2 \ F 1 | ≥ | V ( H )| and | F 1 \ F 2 | ≥ | V ( H )| when F 1 \ F 2 = ∅. Therefore, F 1 ∩ F 2 is also a g-good-neighbor faulty set of G. Note that F 1 ∩ F 2 = F 1 is also a g-extra faulty set when F 1 \ F 2 = ∅. Since there are no edges between V (G − F 1 − F 2 ) and F 1 F 2 , F 1 ∩ F 2 is a g-good-neighbor cut of G. If F 1 ∩ F 2 = ∅, this is a contradiction to that G is connected. Therefore, F 1 ∩ F 2 = ∅ and hence | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ | V ( H )| + κ ( g ) (G ), which contradicts | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 1. So G is g-good-neighbor (κ ( g ) (G ) + | V ( H )| − 1)-diagnosable. By the definition of t g (G ), t g (G ) ≥ κ ( g ) (G ) + | V ( H )| − 1. 2 By Theorems 4.1 and 4.2, we have the following theorem. Theorem 4.3. Let G = ( V (G ), E (G )) be a g-good-neighbor connected graph, and let H be connected subgraph of G δ(G ) = g such that it contains V (G ) as least as possible and N ( V ( H )) is a minimum g-good-neighbor cut of G, and let H be connected subgraph of G with δ(G ) = g such that it contains V (G ) as least as possible. If V (G ) = F 1 ∪ F 2 for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 1 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 1, then κ ( g ) (G ) + | V ( H )| − 1 ≤ t g (G ) ≤ κ ( g ) (G ) + | V ( H )| − 1 under the PMC model.
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Lemma 4.4. ([15]) Let B S n be the bubble-sort star graph and A = {(1), (12), (123), (13)}. If n ≥ 5, F 1 = N ( A ), F 2 = A ∪ N ( A ), then | F 1 | = 8n − 22, | F 2 | = 8n − 18, δ( B S n − F 1 ) ≥ 2, and δ( B S n − F 2 ) ≥ 2. Theorem 4.5. ([15]) For n ≥ 5, the 2-good-neighbor connectivity of the bubble-sort star graph B S n is 8n − 22. By Lemma 4.4, there is connected subgraph B S n [ A ] of minimum degree 2 such that it contains V ( B S n ) as least as possible and N ( A ) is a minimum 2-good-neighbor cut of B S n By Theorem 4.5, κ (2) ( B S n ) = 8n − 22. Since n! > [(8n − 22) + 4 − 1] + [(8n − 22) + 4 − 1] when n ≥ 5, we have V ( B S n ) = F 1 ∪ F 2 for each distinct pair of 2-good-neighbor faulty subsets F 1 and F 2 of B S n with | F 1 | ≤ (8n − 22) + 4 − 1 and | F 2 | ≤ (8n − 22) + 4 − 1. By Theorem 4.3, we have the following corollary. Corollary 4.6. ([15]) For n ≥ 5, the 2-good-neighbor diagnosability of the bubble-sort star graph B S n is 8n − 19 under the PMC model. Let G = ( V (G ), E (G )) be a g-good-neighbor connected graph. Suppose that H is connected subgraph of G with δ( H ) = g such that it contains V (G ) as least as possible. Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H ∗ be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ) for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 2 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 2. Theorem 4.7. Let G be a g-good-neighbor connected graph, and let V ( H ∗ ) = ∅ for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 2 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 2. Then the g-good-neighbor diagnosability of G is more than or equal to κ ( g ) (G ) + | V ( H )| − 2, i.e., t g (G ) ≥ κ ( g ) (G ) + | V ( H )| − 2 under the MM∗ model. Proof. By the definition of g-good-neighbor diagnosability, it is sufficient to show that G is g-good-neighbor (κ ( g ) (G ) + | V ( H )| − 2)-diagnosable. Suppose, on the contrary, that there are two distinct g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 2 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 2, but the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 3.2. Without loss of generality, suppose that F 2 \ F 1 = ∅. Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H ∗ be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ). Then V ( H ∗ ) = ∅. We consider the following cases. Case 1. g = 0. By the definition of H , | V ( H )| = 1. Note that V ( H ∗ ) = ∅ for each distinct pair of 0-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ (0) (G ) + | V ( H )| − 2 = κ (0) (G ) − 1 and | F 2 | ≤ κ (0) (G ) + | V ( H )| − 2 = κ (0) (G ) − 1 and F 2 \ F 1 = ∅. Since the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition (1) of Theorem 3.2, and any vertex of V ( H ∗ ) is not isolated in H ∗ , we deduce that there is no edge between V ( H ∗ ) and F 1 F 2 . Therefore, F 1 ∩ F 2 is a 0-good-neighbor cut of G. Thus, | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ 1 + κ (0) (G ), which contradicts | F 2 | ≤ κ (0) (G ) − 1. Case 2. g ≥ 1. Claim 1. G − F 1 − F 2 has no isolated vertex. Suppose, on the contrary, that G − F 1 − F 2 has at least one isolated vertex w 1 . Since F 1 is one g-good-neighbor faulty set, there is a vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . Meanwhile, since the vertex set pair ( F 1 , F 2 ) is not satisfied with any one condition in Theorem 3.2, by the condition (3) of Theorem 3.2, there is at most one vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . Thus, there is just a vertex u ∈ F 2 \ F 1 such that u is adjacent to w 1 . So d( w 1 ) = 1 in G [{ w 1 } ∪ ( F 2 \ F 1 )]. Since F 1 is a g-good-neighbor faulty set, this is a contradiction when g ≥ 2. Then F 1 is a 1-good-neighbor faulty set. If F 1 \ F 2 = ∅, then F 1 ⊆ F 2 . Since F 2 is a g-good-neighbor faulty set, every vertex v of G − F 1 − F 2 = G − F 2 has d( v ) ≥ g in G − F 2 . Therefore, G − F 1 − F 2 has no isolated vertex for g ≥ 1. Thus, F 1 \ F 2 = ∅. Similarly, we know that there is just a vertex a ∈ F 1 \ F 2 such that a is adjacent to w 1 and F 2 is a 1-good-neighbor faulty set. Let W ⊆ V (G ) \ ( F 1 ∪ F 2 ) be the set of isolated vertices in G [ V (G ) \ ( F 1 ∪ F 2 )], and let H ∗ be the induced subgraph by the vertex set V (G ) \ ( F 1 ∪ F 2 ∪ W ). Then V ( H ∗ ) = ∅. Since the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition (1) of Theorem 3.2, and any vertex of V ( H ∗ ) is not isolated in H ∗ , we deduce that there is no edge between V ( H ∗ ) and F 1 F 2 . Note F 2 \ F 1 = ∅. If F 1 ∩ F 2 = ∅, then this is a contradiction to that G is connected. Therefore, F 1 ∩ F 2 = ∅. Thus, F 1 ∩ F 2 is a vertex cut of G. Since F 1 is a 1-good-neighbor faulty set of G, we have that every vertex v of H ∗ has d H ∗ ( v ) ≥ 1 and every vertex a of G [ W ∪ ( F 2 \ F 1 )]) has d(a) ≥ 1 in G [ W ∪( F 2 \ F 1 )]). Since F 2 is a 1-good-neighbor faulty set of G, we have that every vertex b of G [ W ∪( F 1 \ F 2 )]) has d(b) ≥ 1 in G [ W ∪ ( F 1 \ F 2 )]). Therefore, every vertex x of G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]) has d(x) ≥ 1 in G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]). Note that G − ( F 1 ∩ F 2 ) has two parts (for convenience): H ∗ and G [ W ∪ ( F 1 \ F 2 ) ∪ ( F 2 \ F 1 )]). Therefore, F 1 ∩ F 2 is a 1-good-neighbor cut of G and hence we have that κ (1) (G ) + 2 − 2 ≥ | F 2 | = | F 1 ∩ F 2 | + | F 2 \ F 1 | ≥ κ (1) (G ) + 1, a contradiction. The proof of Claim 1 is complete.
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Let u ∈ V (G ) \ ( F 1 ∪ F 2 ). By Claim 1, u has at least one neighbor vertex in G − F 1 − F 2 . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 3.2, by the condition (1) of Theorem 3.2, for any pair of adjacent vertices u , w ∈ V (G ) \ ( F 1 ∪ F 2 ), there is no vertex v ∈ F 1 F 2 such that u w ∈ E (G ) and v w ∈ E (G ). It follows that u has no neighbor in F 1 F 2 . Since u is taken arbitrarily, so there is no edge between V (G ) \ ( F 1 ∪ F 2 ) and F 1 F 2 . If F 1 ∩ F 2 = ∅, then this is a contradiction to that G is connected. Therefore, F 1 ∩ F 2 = ∅ and F 1 ∩ F 2 is a cut of G. Since F 2 \ F 1 = ∅ and F 1 is a g-good-neighbor faulty set, we have that every vertex v of G − F 1 − F 2 has d( v ) ≥ g ≥ 1 in G − F 1 − F 2 and every vertex a of G ([ F 2 \ F 1 ]) has d(a) ≥ g ≥ 1 in G ([ F 2 \ F 1 ]). By the definition of H , | F 2 \ F 1 | ≥ | V ( H )|. Suppose that F 1 \ F 2 = ∅. Then F 1 ∩ F 2 = F 1 . Since F 1 is a g-good-neighbor faulty set of G, we have that F 1 ∩ F 2 = F 1 is a g-good-neighbor faulty set of G. Since there is no edge between V (G ) \ ( F 1 ∪ F 2 ) and F 2 \ F 1 , we have that F 1 ∩ F 2 = F 1 is a g-good-neighbor cut of G. Suppose that F 1 \ F 2 = ∅. Similarly, every vertex b of G ([ F 1 \ F 2 ]) has d(b) ≥ g. Note that G − ( F 1 ∩ F 2 ) has three parts (for convenience): H ∗ , G [ F 1 \ F 2 ] and G [ F 2 \ F 1 ]. Therefore, F 1 ∩ F 2 is a g-good-neighbor cut of G and hence | F 2 | = | F 2 \ F 1 | + | F 1 ∩ F 2 | ≥ | V ( H )| + κ ( g ) (G ), which contradicts | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 2. Therefore, G is g-good-neighbor (κ ( g ) (G ) + | V ( H )| − 2)-diagnosable and t g (G ) ≥ κ ( g ) (G ) + | V ( H )| − 2. The proof is complete. 2 By Theorems 4.1 and 4.7, we have the following theorem. Theorem 4.8. Let G be a g-good-neighbor connected graph, and let H be connected subgraph of G with δ( H ) = g such that it contains V (G ) as least as possible, and N ( V ( H )) is a minimum g-good-neighbor cut of G, and let H be connected subgraph of G with δ(G ) = g such that it contains V (G ) as least as possible. If V ( H ∗ ) = ∅ for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of G with | F 1 | ≤ κ ( g ) (G ) + | V ( H )| − 2 and | F 2 | ≤ κ ( g ) (G ) + | V ( H )| − 2, then κ ( g ) (G ) + | V ( H )| − 2 ≤ t g (G ) ≤ κ ( g ) (G ) + | V ( H )| − 1 under the MM∗ model. 5. Conclusions Conditional connectivity and conditional diagnosability are two important metrics for fault tolerance of a multiprocessor system. In this paper, we show the relationship between the g-good-neighbor (extra) diagnosability and g-good-neighbor (extra) connectivity of networks. It provides a simple way to find the g-good-neighbor (extra) diagnosability of some wellknown networks based on the g-good-neighbor (extra) connectivity. The work will help engineers to develop more different networks. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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