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The diagnosability of a multiprocessor system or an interconnection network is an important research topic. The system and an interconnection network have an underlying topology, which is usually presented by a graph. In this paper, we show proof for the g-good-neighbor diagnosability of the exchanged hypercube EH (s,t) under the PMC model and MM* model.

A multiprocessor system and interconnection network have an underlying topology, which is usually presented by a graph, where nodes represent processors and links represent communication links between processors. Some processors may fail in the system and processor fault identification plays an important role in reliable computing. The identification process is called the diagnosis of the system. Several diagnosis models were proposed to identify the faulty processors. One major approach is the Preparata, Metze, and Chien’s (PMC) diagnosis model introduced by Preparata et al. [^{*} is a special case of the MM model and each node must test all pairs of its adjacent nodes of the system. The diagnosability of the system is one important study topic. In 2012, Peng et al. [^{*} model, see [

Let E H ( s , t ) be the exchanged hypercube with 1 ≤ s ≤ t . In this paper, we show the following: 1) The g-good-neighbor diagnosability of E H ( s , t ) is 2 g ( s + 2 − g ) − 1 under the PMC model for any g with 0 ≤ g ≤ s . 2) The diagnosability of E H ( s , t ) under the MM^{*} model is s + 1 for 2 ≤ s ≤ t . 3) The g-good-neighbor diagnosability of E H ( s , t ) under the MM^{*} model is 2 g ( s + 2 − g ) − 1 for 3 ≤ s ≤ t and any g with 0 ≤ g ≤ s .

The rest of this paper is organized as follows: In Section 2, we provide the terminology and preliminaries for the system diagnosis. In Section 3, we shall show the g-good-neighbor diagnosability of the exchanged hypercube under the PMC model and the MM^{*} model. Finally, the conclusion is given in Section 4.

A multiprocessor system and a network are modeled as an undirected simple graph G = ( V ( G ) , E ( G ) ) , V ( G ) denotes processors and E ( G ) denotes communication links. For V ′ ⊆ V ( G ) with V ′ ≠ ∅ , the subgraph of G induced by V ′ , denoted by G [ V ′ ] . For F 1 , F 2 ⊆ V ( G ) with F 1 ≠ F 2 , the symmetric difference F 1 Δ F 2 is ( F 1 ∖ F 2 ) ∪ ( F 2 ∖ F 1 ) . For v ∈ V ( G ) , the neighborhood N G ( v ) of v in G to be the set of vertices adjacent to v. Let S ⊆ V ( G ) . The set ∪ v ∈ S N G ( v ) \ S is denoted by N G ( S ) . For graph-theoretical terminology and notation not defined here we follow [

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.

Definition 2.1. A system G = ( V , E ) is g-good-neighbor t-diagnosable under the PMC model if and only if ( F 1 , F 2 ) is distinguishable for each distinct pair of g-good-neighbor 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 under the PMC model. In particular, t 0 ( G ) = t ( G ) is said to be the diagnosability of G under the PMC model, t 1 ( G ) is said to be the nature diagnosability of G under the PMC model.

Definition 2.2. A system G = ( V , E ) is g-good-neighbor t-diagnosable under the MM^{*} model if and only if ( F 1 , F 2 ) is distinguishable for each distinct pair of g-good-neighbor 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 under the MM^{*} model. In particular, t 0 ( G ) = t ( G ) is said to be the diagnosability of G under the MM^{*} model, t 1 ( G ) is said to be the nature diagnosability of G under the MM^{*} model.

For a given position integer n, let [ n ] = { 1 , 2 , ⋯ , n } . The sequence x n x n − 1 ⋯ x 1 is called a binary string of length n if x r ∈ { 0 , 1 } for each r ∈ [ n ] . Let x = x n x n − 1 ⋯ x 1 and y = y n y n − 1 ⋯ y 1 be two distinct binary strings of length n.

Hamming distance between x and y, denoted by H ( x , y ) , is the number of r’s for which | x r − y r | = 1 for r ∈ [ n ] .

For a binary string u = u n u n − 1 ⋯ u 1 u 0 of length n + 1 , we call u r the r-th bit of u for r ∈ [ n ] , and u 0 the last bit of u, denote sub-sequence u j u j − 1 ⋯ u i + 1 u i of u by u [ j : i ] , i.e., u [ j : i ] = u j u j − 1 ⋯ u i + 1 u i . Let V ( s , t ) = { u s + t ⋯ u t + 1 u t ⋯ u 1 u 0 : u 0 , u i ∈ { 0 , 1 } foreach i ∈ [ s + t ] } .

Definition 2.3. The exchanged hypercube is an undirected graph E H ( s , t ) = ( V , E ) , where s ≥ 1 and t ≥ 1 are integers. The set of vertices V is V ( s , t ) , and the set of edges E is composed of three disjoint types E 1 , E 2 and E 3 : E 1 = { u v ∈ V × V : u [ s + t : 1 ] = v [ s + t : 1 ] , u 0 ≠ v 0 } , E 2 = { u v ∈ V × V : u [ s + t : t + 1 ] = v [ s + t : t + 1 ] , H ( u [ t : 1 ] , v [ t : 1 ] ) = 1 , u 0 = v 0 = 1 } , E 3 = { u v ∈ V × V : u [ t : 1 ] = v [ t : 1 ] , H ( u [ s + t : t + 1 ] , v ) [ s + t : t + 1 ] = 1 , u 0 = v 0 = 0 } .

Theorem 3.1. [

Let v 0 = 0 n = 00 ⋯ 0 ︸ n and let V g = { 0 s − g u g + t ⋯ u 1 + t 0 t + 1 : u i = 0 , 1 for i = t + 1 , t + 2 , ⋯ , g + t } . Then E H ( s , t ) [ V g ] ≅ Q g . By the proof of Lemma 3.1 in [

Lemma 3.2. Let E H ( s , t ) be the exchanged hypercube with 1 ≤ s ≤ t . V g is defined as above for 0 ≤ g ≤ s . Then | V g | = 2 g , | N E H ( s , t ) ( V g ) | = 2 g ( s + 1 − g ) , and N E H ( s , t ) ( V g ) is a g-good-neighbor cut of E H ( s , t ) .

Theorem 3.3. [

Theorem 3.4. Let E H ( s , t ) be the exchanged hypercube with 1 ≤ s ≤ t and any g with 0 ≤ g ≤ s . Then the g-good-neighbor diagnosability of E H ( s , t ) is 2 g ( s + 2 − g ) − 1 under the PMC model.

Proof. Let V g be defined in Lemma 3.2 for 0 ≤ g ≤ s . By the definition of E H ( s , t ) [ V g ] ≅ Q g , | V ( Q g ) | is minimum such that δ ( Q g ) = g . Note 2 s + t + 1 > 2 ( 2 g ( s + 1 − g ) + 2 g − 1 ) . Therefore, V ( E H ( s , t ) ) ≠ F 1 ∪ F 2 for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of E H ( s , t ) with | F 1 | ≤ 2 g ( s + 1 − g ) + 2 g − 1 and | F 2 | ≤ 2 g ( s + 1 − g ) + 2 g − 1 . By Theorem 3.4, t g ( E H ( s , t ) ) ≥ 2 g ( s + 1 − g ) + 2 g − 1 . On the other hand, by Lemma 3.2, N E H ( s , t ) ( V g ) is a g-good-neighbor cut of E H ( s , t ) . Since | N E H ( s , t ) ( V g ) | = 2 g ( s + 1 − g ) , by Theorem 3.1, N E H ( s , t ) ( V g ) is a minimum g-good-neighbor cut of E H ( s , t ) . Note | V g | = 2 g . By Theorem 3.4, t g ( E H ( s , t ) ) ≤ 2 g ( s + 1 − g ) + 2 g − 1 . Therefore, t g ( E H ( s , t ) ) = 2 g ( s + 1 − g ) + 2 g − 1 . ,

Before discussing the g-good-neighbor diagnosability of the exchanged hypercube under the MM^{*} model, we first give two existing results.

Theorem 3.5. [^{*} model if and only if for each distinct pair of 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. 1) There are two vertices u , w ∈ V ∖ ( F 1 ∪ F 2 ) and there is a vertex v ∈ F 1 Δ 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 .

Theorem 3.6. [^{*} model.

Lemma 3.7. Let E H ( s , t ) be the exchanged hypercube with 1 ≤ s ≤ t and any g with 0 ≤ g ≤ s . Then the g-good-neighbor diagnosability of the exchanged hypercube E H ( s , t ) under the MM^{*} model is less than or equal to 2 g ( s + 1 − g ) + 2 g − 1 , i.e., t g ( E H ( s , t ) ) ≤ 2 g ( s + 2 − g ) − 1 .

Proof. Let V g be defined in Lemma 3.2 for 0 ≤ g ≤ s . By the definition of E H ( s , t ) [ V g ] ≅ Q g , | V ( Q g ) | = 2 g is minimum such that δ ( Q g ) = g . By Lemma 3.2, N E H ( s , t ) ( V g ) is a g-good-neighbor cut of E H ( s , t ) . By Theorems 3.6 and 3.1, t g ( G ) ≤ κ ( g ) ( G ) + | V ( H ) | − 1 = 2 g ( s + 2 − g ) − 1 . ,

A component of a graph G is odd according as it has an odd number of vertices. We denote by o ( G ) the number of odd component of G.

Lemma 3.8. [

Lemma 3.9. [^{*} model.

Theorem 3.10. Let E H ( s , t ) be the exchanged hypercube with 2 ≤ s ≤ t . Then the 0-good-neighbor diagnosability of E H ( s , t ) under the MM^{*} model is s + 1 , i.e.， t 0 ( E H ( s , t ) ) = t ( E H ( s , t ) ) = s + 1 .

Proof. By the definition of the g-good-neighbor diagnosability, it is sufficient to show that E H ( s , t ) is 0-good-neighbor ( s + 1 ) -diagnosable.

By Theorem 3.5, suppose, on the contrary, that there are two distinct 0-good-neighbor faulty subsets F 1 and F 2 of E H ( s , t ) with | F 1 | ≤ s + 1 and | F 2 | ≤ s + 1 , but the vertex set pair ( F 1 , F 2 ) is not satisfied with any one condition in Theorem 3.5. Without loss of generality, assume that F 2 ∖ F 1 ≠ ∅ . Note | F 1 ∪ F 2 | ≤ | F 1 | + | F 2 | ≤ 2 s + 2 < 2 s + t + 1 . Therefore, V ( E H ( s , t ) ) ≠ F 1 ∪ F 2 .

Note E H ( s , t ) has a perfect matching. Let W ⊆ V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) be the set of isolated vertices in E H ( s , t ) [ V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) ] , and let H be the subgraph induced by the vertex set V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ∪ W ) . By Lemma 3.8, | W | ≤ o ( E H ( s , t ) − ( F 1 ∪ F 2 ) ) ≤ | F 1 ∪ F 2 | . Note 2 | F 1 ∪ F 2 | ≤ 2 ( 2 s + 2 ) < 2 s + t + 1 = | V ( E H ( s , t ) ) | . Therefore, V ( H ) ≠ ∅ . Since F 1 and F 2 are two distinct 0-good-neighbor faulty sets, and there is no edge between V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) and F 1 Δ F 2 , we have that F 1 ∩ F 2 is a 0-good-neighbor cut of E H ( s , t ) . By Theorem 3.1, we have | F 1 ∩ F 2 | ≥ s + 1 . Therefore, | F 2 | = | F 2 ∖ F 1 | + | F 1 ∩ F 2 | ≥ 1 + s + 1 , which contradicts | F 2 | ≤ s + 1 . Therefore, E H ( s , t ) is 0-good-neighbor ( s + 1 ) -diagnosable and t 0 ( E H ( s , t ) ) ≥ s + 1 . Combining this with Lemma 3.9, we have t 0 ( E H ( s , t ) ) = t ( E H ( s , t ) ) = s + 1 . ,

Theorem 3.11. Let E H ( s , t ) be the exchanged hypercube with 3 ≤ s ≤ t . Then the 1-good-neighbor diagnosability of E H ( s , t ) under the MM^{*} model is 2 s + 1 , i.e., t 1 ( E H ( s , t ) ) = 2 s + 1 .

Proof. By the definition of 1-good-neighbor diagnosability, it is sufficient to show that E H ( s , t ) is 1-good-neighbor ( 2 s + 1 ) -diagnosable.

By Theorem 3.5, suppose, on the contrary, that there are two distinct 1-good-neighbor faulty subsets F 1 and F 2 of E H ( s , t ) with | F 1 | ≤ 2 s + 1 and | F 2 | ≤ 2 s + 1 , but the vertex set pair ( F 1 , F 2 ) is not satisfied with any one condition in Theorem 3.5. Without loss of generality, assume that F 2 ∖ F 1 ≠ ∅ . Note | F 1 ∪ F 2 | ≤ | F 1 | + | F 2 | ≤ 4 s + 2 < 2 s + t + 1 . Therefore, V ( E H ( s , t ) ) ≠ F 1 ∪ F 2 .

Claim I. E H ( s , t ) − F 1 − F 2 has no isolated vertex.

Suppose, on the contrary, that E H ( s , t ) − F 1 − F 2 has at least one isolated vertex w. Since F 1 is a 1-good neighbor faulty set, there is a vertex u ∈ F 2 ∖ F 1 such that u is adjacent to w. Since the vertex set pair ( F 1 , F 2 ) is not satisfied with any one condition in Theorem 3.5, there is at most one vertex u ∈ F 2 ∖ F 1 such that u is adjacent to w. Thus, there is just a vertex u ∈ F 2 ∖ F 1 such that u is adjacent to w. If F 1 ∖ F 2 = ∅ , then F 1 ⊆ F 2 . Since F 2 is a 1-good neighbor faulty set, E H ( s , t ) − F 2 = E H ( s , t ) − F 1 − F 2 has not any isolated vertex; a contradiction. Therefore, F 1 ∖ F 2 ≠ ∅ . Similarly, we can deduce that there is just a vertex v ∈ F 1 ∖ F 2 such that v is adjacent to w. Let W ⊆ V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) be the set of isolated vertices in E H ( s , t ) [ V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) ] , and let H be the subgraph induced by the vertex set V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ∪ W ) . Then for any w ∈ W , there are ( s − 1 ) neighbors in F 1 ∩ F 2 . Note E H ( s , t ) has a perfect matching. By Lemma 3.8, | W | ≤ o ( E H ( s , t ) − ( F 1 ∪ F 2 ) ) ≤ | F 1 ∪ F 2 | = | F 1 | + | F 2 | − | F 1 ∩ F 2 | ≤ 2 ( 2 s + 1 ) − ( s − 1 ) = 3 s + 3 . Suppose V ( H ) = ∅ . Then 2 s + t + 1 = | V ( E H ( s , t ) ) | = | F 1 ∪ F 2 | + | W | ≤ 6 s + 6 . This is a contradiction to s ≥ 3 . So V ( H ) ≠ ∅ . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition (1) of Theorem 3.5, and any vertex of V ( H ) is not isolated in H, we induce that there is no edge between V ( H ) and F 1 Δ F 2 . Thus, F 1 ∩ F 2 is a vertex cut of E H ( s , t ) and δ ( E H ( s , t ) − ( F 1 ∩ F 2 ) ) ≥ 1 , i.e., F 1 ∩ F 2 is a 1-good-neighbor cut of E H ( s , t ) . By Theorem 3.1, | F 1 ∩ F 2 | ≥ 2 s . Because | F 1 | ≤ 2 s + 1 , | F 2 | ≤ 2 s + 1 , and neither F 1 ∖ F 2 nor F 2 ∖ F 1 is empty, we have | F 1 ∖ F 2 | = | F 2 ∖ F 1 | = 1 and | F 2 ∩ F 1 | = 2 s . Let F 1 ∖ F 2 = { v 1 } and F 2 ∖ F 1 = { v 2 } . Then for any vertex w ∈ W , w are adjacent to v 1 and v 2 . Note that there are at most two common neighbors for any pair of vertices in E H ( s , t ) , it follows that there are at most two isolated vertices in E H ( s , t ) − F 1 − F 2 .

Suppose that there is exactly one isolated vertex v in E H ( s , t ) − F 1 − F 2 . Let v 1 and v 2 be adjacent to v. Then N E H ( s , t ) ( v ) ∖ { v 1 , v 2 } ⊆ F 1 ∩ F 2 . Since E H ( s , t ) contains no triangle, it follows that N E H ( s , t ) ( v 1 ) ∖ { v } ⊆ F 1 ∩ F 2 ; N E H ( s , t ) ( v 2 ) ∖ { v } ⊆ F 1 ∩ F 2 ; [ N E H ( s , t ) ( v ) ∖ { v 1 , v 2 } ] ∩ [ N E H ( s , t ) ( v 1 ) ∖ { v } ] = ∅ , [ N E H ( s , t ) ( v ) ∖ { v 1 , v 2 } ] ∩ [ N E H ( s , t ) ( v 2 ) ∖ { v } ] = ∅ and | [ N E H ( s , t ) ( v 1 ) ∖ { v } ] ∩ [ N E H ( s , t ) ( v 2 ) ∖ { v } ] | ≤ 1 .

Thus, | F 1 ∩ F 2 | ≥ | N E H ( s , t ) ( v ) ∖ { v 1 , v 2 } | + | N E H ( s , t ) ( v 1 ) ∖ { v } | + | N E H ( s , t ) ( v 2 ) ∖ { v } | ≥ ( s − 1 ) + s + s − 1 ≥ 3 s − 2 . It follows that | F 2 | = | F 2 ∖ F 1 | + | F 1 ∩ F 2 | ≥ 1 + 3 s − 2 = 3 s − 1 > 2 s + 1 ( s ≥ 3 ) , which contradicts | F 2 | ≤ 2 s + 1 .

Suppose that there are exactly two isolated vertices v and w in E H ( s , t ) − F 1 − F 2 . Let v 1 and v 2 be adjacent to v and w, respectively. Then N E H ( s , t ) ( v ) ∖ { v 1 , v 2 } ⊆ F 1 ∩ F 2 . Since E H ( s , t ) contains no triangle, it follows that N E H ( s , t ) ( v 1 ) ∖ { v , w } ⊆ F 1 ∩ F 2 , N E H ( s , t ) ( v 2 ) ∖ { v , w } ⊆ F 1 ∩ F 2 , [ N E H ( s , t ) ( v ) ∖ { v 1 , v 2 } ] ∩ [ N E H ( s , t ) ( v 1 ) ∖ { v , w } ] = ∅ , [ N E H ( s , t ) ( v ) ∖ { v 1 , v 2 } ] ∩ [ N E H ( s , t ) ( v 2 ) ∖ { v , w } ] = ∅ and | [ N E H ( s , t ) ( v 1 ) ∖ { v , w } ] ∩ [ N E H ( s , t ) ( v 2 ) ∖ { v , w } ] | = 0 .

Thus, | F 1 ∩ F 2 | ≥ | N E H ( s , t ) ( v ) ∖ { v 1 , v 2 } | + | N E H ( s , t ) ( w ) ∖ { v 1 , v 2 } | + | N E H ( s , t ) ( v 1 ) ∖ { v , w } | + | N E H ( s , t ) ( v 2 ) ∖ { v , w } | = ( s − 1 ) + ( s − 1 ) + ( s − 1 ) + ( s − 1 ) = 4 s − 4 . It follows that | F 2 | = | F 2 ∖ F 1 | + | F 1 ∩ F 2 | ≥ 1 + 4 s − 4 = 4 s − 3 > 2 s + 1 ( s ≥ 3 ) , which contradicts | F 2 | ≤ 2 s + 1 . The proof of Claim I is complete.

Let u ∈ V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) . By Claim I, u has at least one neighbor in E H ( s , t ) − F 1 − F 2 . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with any one condition in Theorem 3.5, by the condition (1) of Theorem 3.5, for any pair of adjacent vertices u , w ∈ V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) , there is no vertex v ∈ F 1 Δ F 2 such that u w ∈ E ( E H ( s , t ) ) and v w ∈ E ( E H ( s , t ) ) . It follows that u has no neighbor in F 1 Δ F 2 . By the arbitrariness of u, there is no edge between V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) and F 1 Δ F 2 . Since F 2 ∖ F 1 ≠ ∅ and F 1 is a 1-good-neighbor faulty set, δ E H ( s , t ) ( [ F 2 ∖ F 1 ] ) ≥ 1 . Note | F 2 ∖ F 1 | ≥ 2 . Since both F 1 and F 2 are 1-good-neighbor faulty sets, and there is no edge between V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) and F 1 Δ F 2 , F 1 ∩ F 2 is a 1-good-neighbor cut of E H ( s , t ) . By Theorem 3.1, | F 1 ∩ F 2 | ≥ 2 s . Therefore, | F 2 | = | F 2 ∖ F 1 | + | F 1 ∩ F 2 | ≥ 2 + 2 s = 2 s + 2 , which contradicts | F 2 | ≤ 2 s + 1 . Therefore, E H ( s , t ) is 1-good-neighbor ( 2 s + 1 ) -diagnosable and t 1 ( E H ( s , t ) ) ≥ 2 s + 1 . Combining this with Lemma 3.7, we have t 1 ( E H ( s , t ) ) = 2 s + 1 . ,

Theorem 3.12. Let E H ( s , t ) be the exchanged hypercube with 3 ≤ s ≤ t and any g with 0 ≤ g ≤ s . Then the g-good-neighbor diagnosability of the exchanged hypercube E H ( s , t ) under the MM^{*} model is 2 g ( s + 1 − g ) + 2 g − 1 , i.e., t g ( E H ( s , t ) ) = 2 g ( s + 2 − g ) − 1 .

Proof. By the definition of the g -good-neighbor diagnosability, it is sufficient to show that E H ( s , t ) is g-good-neighbor ( 2 g ( s + 1 − g ) + 2 g − 1 ) -diagnosable. By Theorems 3.10 and 3.11, it is sufficient to show that g ≥ 2 .

By Theorem 3.5, suppose, on the contrary, that there are two distinct g-good-neighbor faulty subsets F 1 and F 2 of E H ( s , t ) with | F 1 | ≤ 2 g ( s + 1 − g ) + 2 g − 1 and | F 2 | ≤ 2 g ( s + 1 − g ) + 2 g − 1 , but the vertex set pair ( F 1 , F 2 ) is not satisfied with any one condition in Theorem 3.5. Without loss of generality, assume that F 2 ∖ F 1 ≠ ∅ . It is easy to verify | V ( E H ( s , t ) ) | = 2 s + t + 1 > 2 ( 2 g ( s + 1 − g ) + 2 g − 1 ) = | F 1 ∪ F 2 | . Therefore, V ( E H ( s , t ) ) ≠ F 1 ∪ F 2 .

Claim 1. E H ( s , t ) − F 1 − F 2 has no isolated vertex.

Suppose, on the contrary, that E H ( s , t ) − F 1 − F 2 has at least one isolated vertex x. Since F 1 is a g-good neighbor faulty set and g ≥ 2 , there are at least two vertices u , v ∈ F 2 ∖ F 1 such that u , v are adjacent to x. According to the hypothesis, the vertex set pair ( F 1 , F 2 ) is not satisfied with any one condition in Theorem 3.5. By the condition (3) of Theorem 3.5, there are at most one vertex u ∈ F 2 ∖ F 1 such that u are adjacent to x. So | N E H ( s , t ) − F 1 ( x ) | ≤ 1 , a contradiction to that F 1 is a g-good neighbor faulty set, where g ≥ 2 . Thus, E H ( s , t ) − F 1 − F 2 has no isolated vertex.

The proof of Claim 1 is complete.

Let u ∈ V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) . By Claim 1, δ ( E H ( s , t ) − F 1 − F 2 ) ≥ 1 . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with any one condition in Theorem 3.5, by the condition (1) of Theorem 3.5, for any pair of adjacent vertices u , w ∈ V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) , there is no vertex v ∈ F 1 Δ F 2 such that u w ∈ E ( E H ( s , t ) ) and u v ∈ E ( E H ( s , t ) ) . It follows that u has no neighbor in F 1 Δ F 2 . By the arbitrariness of u, there is no edge between V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) and F 1 Δ F 2 . Since F 2 ∖ F 1 ≠ ∅ and F 1 is a g-good-neighbor faulty set, δ E H ( s , t ) ( [ F 2 ∖ F 1 ] ) ≥ g and δ ( E H ( s , t ) − F 2 − F 1 ) ≥ g . By the definition of E H ( s , t ) , | F 2 ∖ F 1 | ≥ 2 g . Since both F 1 and F 2 are g-good-neighbor faulty sets, and there is no edge between V ( E H ( s , t ) ) ∖ ( F 1 ∪ F 2 ) and F 1 Δ F 2 , F 1 ∩ F 2 is a g-good-neighbor cut of E H ( s , t ) . By Theorem 3.1, | F 1 ∩ F 2 | ≥ 2 g ( s + 1 − g ) . Therefore, | F 2 | = | F 2 ∖ F 1 | + | F 1 ∩ F 2 | ≥ 2 g + 2 g ( s + 1 − g ) , which contradicts | F 2 | ≤ 2 g ( s + 1 − g ) + 2 g − 1 . Therefore, E H ( s , t ) is g-good-neighbor ( 2 g ( s + 1 − g ) + 2 g − 1 ) -diagnosable and t g ( E H ( s , t ) ) ≥ 2 g ( s + 1 − g ) + 2 g − 1 . Combining this with Lemma 3.7, we have t g ( E H ( s , t ) ) = 2 g ( s + 1 − g ) + 2 g − 1 . ,

In this paper, we investigate the problem of the diagnosability of the exchanged hypercube E H ( s , t ) . We show the following. Let E H ( s , t ) be the exchanged hypercube with 3 ≤ s ≤ t and any g with 0 ≤ g ≤ s . Then the g-good-neighbor diagnosability of E H ( s , t ) under the PMC model and MM^{*} model is 2 g ( s + 2 − g ) − 1 . The work will help engineers to develop more different measures of the diagnosability based on application environment, network topology, network reliability, and statistics related to fault patterns.

This work is supported by the National Natural Science Foundation of China (61772010) and the Science Foundation of Henan Normal University (Xiao 20180529, 20180454).

The authors declare no conflicts of interest regarding the publication of this paper.

Ren, Y.X. and Wang, S.Y. (2019) A Proof for g-Good-Neighbor Diagnosability of Exchanged Hypercubes. American Journal of Computational Mathematics, 9, 133-142. https://doi.org/10.4236/ajcm.2019.93010