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g-Good-Neighbor Diagnosability of Arrangement Graphs under the PMC Model and MM* Model

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, Henan, China
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Author to whom correspondence should be addressed.
Information 2018, 9(11), 275; https://doi.org/10.3390/info9110275
Received: 17 September 2018 / Revised: 27 October 2018 / Accepted: 5 November 2018 / Published: 7 November 2018
(This article belongs to the Special Issue Graphs for Smart Communications Systems)

Abstract

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. Under the PMC model, to diagnose the system, two adjacent nodes in G are can perform tests on each other. Under the MM model, to diagnose the 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. As a famous topology structure, the ( n , k ) -arrangement graph A n , k , has many good properties. In this paper, we give the g-good-neighbor diagnosability of A n , k under the PMC model and MM* model.
Keywords: interconnection network; diagnosability; arrangement graph interconnection network; diagnosability; arrangement graph

1. Introduction

A multiprocessor system and interconnection network (networks for short) has 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, 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 [1,2,3]. For a t-diagnosable system, Dahbura and Masson [1] proposed an algorithm with time complex O ( n 2 . 5 ) , which can effectively identify the set of faulty processors.
Several diagnosis models were proposed to identify the faulty processors. One of most commonly used is the Preparata, Metze, and Chien’s (PMC) diagnosis model introduced by Preparata et al. [4]. The diagnosis of the system is achieved through two linked processors testing each other. A similar issue, namely the comparison diagnosis model (MM model), was proposed by Maeng and Malek [5]. In the MM model, to diagnose the 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. [3] introduced a measurement for fault diagnosis of a system, namely, the conditional diagnosability. They considered the situation that no fault set can contain all the neighbors of any vertex in the system. In 2012, Peng et al. [6] proposed a measurement for fault diagnosis of the system G, namely, the g-good-neighbor diagnosability t g ( G ) (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 [6], they studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the PMC model. In [7], Wang and Han studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the MM* model. There is a significant amount of research on the g-good-neighbor diagnosability of graphs [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].
The star graph, which was proposed by Akers et al. [25], is a well-known interconnection network. To solve the problem of scalability of star graph topology, Day and Tripathi [26] proposed the arrangement graph as a generalization of the star graph. The arrangement graph A n , k is more flexible than the star graph in selecting the major design parameters: the number, degree, and diameter of the vertex. At the same time, most of the nice properties of the star graph are preserved (for details, see [26,27,28,29,30,31,32]). In this paper, we show (1) ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 t g ( A n , k ) [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) under the PMC model and MM* model for n 4 , k [ 3 , n 2 ] , g [ 3 , n k ) ; (2) the diagnosability t ( A n , k ) = k ( n k ) under the PMC model and MM* model; (3) t 1 ( A n , k ) = ( 2 k 1 ) ( n k ) under the PMC model for n 5 and k [ 2 , n ) , and under the MM * model for n 8 and k [ 2 , n ) ; (4) t 2 ( A n , k ) = ( 3 k 2 ) ( n k ) under the PMC model and MM* model for n 8 and k [ 3 , n 5 ] { n 2 , n 1 } ; and (5) t 2 ( A n , 2 ) = 4 n 9 under the PMC model and MM* model for n 8 .

2. Preliminaries

Under the PMC model [5,23], to diagnose a system G = ( V ( G ) , E ( G ) ) , two adjacent nodes in G can 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 pair ( u , v ) . The outcome of a test ( u , v ) is 1 (respectively, 0) if u evaluate v as faulty (respectively, fault-free). We assume that the test result is reliable (respectively, unreliable) if the node u is fault-free (respectively, 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.
In the MM model, a processor sends the same task to a pair of distinct neighbors and then compares their responses to diagnose a system G. The comparison scheme of 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 ) . We usually assume that the testing result is reliable (respectively, unreliable) if the node u is fault-free (respectively, faulty). If u , v F and w V ( G ) \ F , then ( u , v ) w 1 . If u F and v , w V ( G ) \ F , then ( u , v ) w 1 . If v F and u , w V ( G ) \ F , then ( u , v ) w 1 . If u , v , w V ( G ) \ F , then ( u , v ) w 0 . The collection of all comparison results in M = ( V ( G ) , L ) is called the syndrome of the diagnosis, denoted by σ . 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* is a special case of the MM model and each node must test its any pair of adjacent nodes, i.e., if u w , v w E ( G ) , then ( u , v ) w L . The set of all faulty processors in the system is called a faulty set. This can be any subset of V ( G ) . For a given syndrome σ , a faulty 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 ) w L such that w V \ F , σ ( u , v ) w = 1 if and only if u , v F or u F or v F under the MM * model. Let σ ( F ) denote the set of all syndromes which F is consistent with. Let F 1 and F 2 be two distinct faulty sets in V ( G ) . If σ ( F 1 ) σ ( F 2 ) , we say ( F 1 , F 2 ) is an indistinguishable pair under the MM * model; else, ( F 1 , F 2 ) is a distinguishable pair under the MM * model.
Definition 1.
A system G = ( V , E ) is g-good-neighbor t-diagnosable if F 1 and F 2 are distinguishable under the PMC (MM * ) model 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 (MM * ) model.
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 . 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. We denote the numbers of vertices and edges in G by | V ( G ) | and | E ( G ) | . The degree d G ( v ) of a vertex v is the number of neighbors of v in G. The minimum degree of a vertex in G is denoted by δ ( G ) . Let S V . We use N G ( S ) to denote the set v S N G ( v ) \ S . For neighborhoods and degrees, we usually omit the subscript for the graph when no confusion arises. A path in G is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. The path with a length of n is denoted by n-path. The length of a shortest path between x and y is called the distance between x and y, denoted by d G ( x , y ) . A complete graph K n is a graph in which any two vertices are adjacent on n vertices. A graph G 1 is isomorphic to another graph G 2 (denoted by G 1 G 2 ) if and only if there exists a bijection φ : V ( G 1 ) V ( G 2 ) such that for any two vertices u , v V ( G 1 ) , u v E ( G 1 ) if and only if φ ( u ) φ ( v ) E ( G 2 ) . 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 [33].
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.
For two positive integers n and k, let n denote the set { 1 , 2 , , n } and k denote the set { 1 , 2 , , k } . Let P n , k be a set of arrangements of k elements in n , that is, P n , k = { p 1 p 2 p k : p i n for 1 i k and p s p t for 1 s , t k , s t } .
Definition 2.
Given two positive integers n and k with n > k 1 . The ( n , k ) -arrangement graph, denoted by A n , k , has vertex set V ( A n , k ) = { p : p = p 1 p k P n , k } , and edge set E ( A n , k ) = { ( p , q ) : p , q V ( A n , k ) with p i q i for some i k and p j = q j for all j k \ { i } } .
From the definition, we know that the vertices of A n , k are the arrangements of k elements in n , and the edges of A n , k connect arrangements which differ in exactly one of their k positions. A n , k is a regular graph of degree k ( n k ) with n ! ( n k ) ! vertices. Figure 1 shows the arrangement graph A 4 , 2 .
Definition 3
([26]). A graph is vertex-transitive if and only if for any pair of its vertices u and v, there exists an automorphism of the graph that maps u to v. A graph is edge-transitive if and only if for any pair of its edges ( u , v ) and ( x , y ) , there exists an automorphism of the graph that maps ( u , v ) to ( x , y ) .
Lemma 1
([26]). A n , k is vertex-transitive and edge-transitive.
Lemma 2
([26]). κ ( A n , k ) = k ( n k ) for n > k 1 .
Lemma 3
([28]). n 3 and n 4 , k [ 2 , n ) , κ ( 1 ) ( A n , k ) = ( 2 k 1 ) ( n k ) 1 and κ ( 1 ) ( A 4 , 2 ) = κ ( 1 ) ( A 4 , 3 ) ( = κ ( 1 ) ( S 4 ) ) = 4 .
Lemma 4
([28]). n 3 and n 4 , 2 k < n , κ ( 1 ) ( A n , k ) = ( 2 k 1 ) ( n k ) 1 and κ ( 1 ) ( A 4 , 2 ) = κ ( 1 ) ( A 4 , 3 ) ( = κ ( 1 ) ( S 4 ) ) = 4 .
Lemma 5
([28]). For n 8 , κ ( 2 ) ( A n , 2 ) = 4 n 12 , and, for k { i : i = 3 , , n 5 } { n 2 , n 1 } , κ ( 2 ) ( A n , k ) = ( 3 k 2 ) ( n k ) 2 .
Lemma 6
([28]). Let n , k , g be positive integers such that n 4 , 2 k n 2 , g 3 . Then,
( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) < κ ( g ) ( A n , k ) [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) g .
An edge cut of a graph G is a set of edges whose removal makes the remaining graph no longer connected. The edge connectivity λ ( G ) of G is the minimum cardinality of an edge cut of G.
Lemma 7
([33]). κ ( G ) λ ( G ) δ ( G ) .
According to Lemmas 2 and 7, we get the following corollary.
Corollary 1.
The edge connectivity λ ( A n , k ) = k ( n k ) for n > k 1 .
For i n , j k , let V ( A n , k j : i ) be the set of all vertices in A n , k with the jth position being i, that is, V ( A n , k j : i ) = { p : p = p 1 p j p k P n , k with p j = i } . It is easy to check that each A n , k j : i is a subgraph of A n , k , and we say that A n , k is decomposed into n subgraphs A n , k j : i ( 1 i n ) according to the jth position. For simplicity, we shall take j as the last position k, and use A n , k i to denote A n , k k : i . Then, V ( A n , k i ) = { p : p = p 1 p k 1 i with p j n \ { i } and p s p t for 1 s , t k 1 } for 1 i n . It is easy to see that A n , 1 is a complete graph K n .
Proposition 1
([34]). Let n > k 2 . For each j k , A n , k j : i is isomorphic to A n 1 , k 1 where 1 i n .
For any vertex u V ( A n , k i ) ( 1 i n ) , in this paper, we say that N ( u ) V ( A n , k i ) is the set of inner neighbors of u, which is denoted by I N ( u ) and N ( u ) ( V ( A n , k ) \ V ( A n , k i ) ) is the set of outer neighbors of u, which is denoted by O N ( u ) .
Proposition 2
([31]). Let n > k 2 , i n . For any two vertices u, v in the subgraph A n , k i , O N ( u ) O N ( v ) = and | O N ( u ) | = n k . Furthermore, the vertices of O N ( u ) are distributed in ( n k ) distinct subgraphs.
Proposition 3.
For any vertex u V ( A n , k i ) ( 1 i n ) , let O N ( u ) be the set of outer neighbors of u. Then, A n , k [ { u } O N ( u ) ] is isomorphic to the complete graph K n k + 1 .
Proof. 
By Lemma 1, A n , k is vertex-transitive. Without loss of generality, let u = ( n k + 1 ) ( n k + 2 ) n V ( A n , k n ) . By the definition of arrangement graphs, O N ( u ) = { u j : u j = ( n k + 1 ) ( n k + 2 ) ( n 1 ) j , j { 1 , 2 , , n k } } . Then, | O N ( u ) | = n k . Note that u, u 1 ,…, u n k 1 and u n k are only different in last position. By the definition of arrangement graphs, any pair of vertices of u u 1 ,…, u n k 1 and u n k are adjacent. Thus, A n , k [ { u } O N ( u ) ] is a complete graph. Note that | { u } O N ( u ) | = n k + 1 . Thus, A n , k [ { u } O N ( u ) ] is isomorphic to K n k + 1 . □
Definition 4.
Let n = { 1 , 2 , , n } , and let S n be the symmetric group on n containing all permutations p = p 1 p 2 p n of n . The alternating group A n is the subgroup of S n containing all even permutations. It is well known that { ( 12 i ) , ( 1 i 2 ) , 3 i n } is a generating set for A n . The n-dimensional alternating group graph A G n is the graph with vertex set V ( A G n ) = A n in which two vertices u, v are adjacent if and only if u = v ( 12 i ) or u = v ( 1 i 2 ) , 3 i n .
Definition 5.
The n-dimensional star graph denoted by S n . The vertex set of S n is { u 1 u 2 u n : u 1 u 2 u n is a permutation of n } . Vertex adjacency is defined as follows: u 1 u 2 u n is adjacent to u i u 2 u i 1 u 1 u i + 1 u n for all 2 i n .
Lemma 8
([29]). (1). The arrangement graph A n , n 2 is isomorphic to the n-dimensional alternating group graph A G n . (2). The arrangement graph A n , n 1 is isomorphic to the n-dimensional star graph S n .
Lemma 9
([31]). For any two distinct vertices u and v in the arrangement graph A n , k , we have the following results:
1. 
If d ( u , v ) = 1 , then | N ( u ) N ( v ) | = n k 1 ;
2. 
If d ( u , v ) = 2 and n = k + 1 , then | N ( u ) N ( v ) | = 1 ;
3. 
If d ( u , v ) = 2 and n k + 2 , then | N ( u ) N ( v ) | 2 ; and
4. 
If d ( u , v ) 3 , then | N ( u ) N ( v ) | = 0 .

3. The g-Good-Neighbor Diagnosability of Arrangement Graphs under the PMC Model

In this section, we show the g-good-neighbor diagnosability of arrangement graphs under the PMC model (Figure 2).
Theorem 1
([23]). A system G = ( V , E ) is g-good-neighbor t-diagnosable under the P M C model if and only if there is an edge u v E with u V \ ( F 1 F 2 ) and v F 1 F 2 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 .
Lemma 10
([28]). For n 3 and n 4 , 2 k < n , κ ( 1 ) ( A n , k ) = ( 2 k 1 ) ( n k ) 1 and κ ( 1 ) ( A 4 , 2 ) = κ ( 1 ) ( A 4 , 3 ) ( = κ ( 1 ) ( S 4 ) ) = 4 .
Lemma 11
([28]). For n 8 , κ ( 2 ) ( A n , 2 ) = 4 n 12 , and, for k { i : i = 3 , , n 5 } { n 2 , n 1 } , κ ( 2 ) ( A n , k ) = ( 3 k 2 ) ( n k ) 2 .
Lemma 12
([27]). Let n 7 and let T be a subset of the vertices of A n , 2 such that | T | 4 n 12 . Then, A n , 2 T is either connected or has a large component and small components with at most two vertices or | T | = 4 n 12 and A n , 2 T has a large component and a four-cycle.
Lemma 13
([28]). Let n , k , g be positive integers such that n 4 , 2 k n 2 , 3 g < n k . Then,
( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) < κ ( g ) ( A n , k ) [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) g .
Let α P ( n , k 1 ) , α = p 1 p k 1 and V α = { p 1 p k 1 i : i n \ { p 1 , , p k 1 } } . Let u = α i = p 1 p k 1 i and v = α j = p 1 p k 1 j , i j and neither i nor j occurs in α . Clearly, u , v V ( A n , k ) , and ( u , v ) E ( A n , k ) . Since any symbol that does not occur in α can serve as the last symbol in a vertex in V α , | V α | = n ( k 1 ) . Thus, the graph K n + k + 1 α induced by V α is a complete graph of order n k + 1 . Let g [ 0 , n k ) and X V ( K n + k + 1 α ) such that | X | = g + 1 . Notice that g + 1 = | X | < | V ( K n + k + 1 α ) | = n k + 1 . Then, A n , k [ X ] is a complete graph K g + 1 .
Lemma 14.
Let n , k , g be positive integers such that n 3 , 2 k < n , 0 g < n k , and let A n , k be the arrangement graph. Let X be defined as above, and let F 1 = N A n , k ( X ) , F 2 = X N A n , k ( X ) . Then, | F 1 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) g , | F 2 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 , δ ( A n , k [ X ] ) g and δ ( A n , k F 1 F 2 ) g .
Proof. 
Let X be defined as above. By the process of the proof of Lemma 13 in [28], N ( X ) is a g-good-neighbor cut of A n , k and | N ( X ) | = | F 1 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) g . Since | X | = g + 1 , | F 2 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 . □
Lemma 15.
Let n 3 , 2 k < n and 0 g < n k . Then, the g-good-neighbor diagnosability of the arrangement graph A n , k under the PMC model is less than or equal to [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) , i.e., t g ( A n , k ) [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) .
Proof. 
Let X be defined as above, and let F 1 = N A n , k ( X ) , F 2 = X N A n , k ( X ) . By Lemma 14, | F 1 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) g , | F 2 | = | X | + | F 1 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 , δ ( A n , k F 1 ) g and δ ( A n , k F 2 ) g . Therefore, F 1 and F 2 are g-good-neighbor faulty sets of A n , k with | F 1 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) g and | F 2 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 .
We prove that A n , k is not g-good-neighbor ( [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 ) -diagnosable. Since X = F 1 F 2 and N A n , k ( X ) = F 1 F 2 , there is no edge of A n , k between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . By Theorem 1, we can show that A n , k is not g-good-neighbor ( [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 ) -diagnosable under the PMC model. Hence, by the definition of the g-good-neighbor diagnosability, we show that the g-good-neighbor diagnosability of A n , k is less than [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 , i.e., t g ( A n , k ) [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) . □
Lemma 16.
Let n , k , g be positive integers such that n 4 , 2 k n 2 , 3 g < n k . Then, the arrangement graph A n , k is g-good-neighbor ( ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 ) -diagnosable under the PMC model.
Proof. 
By Theorem 1, to prove A n , k is g-good-neighbor ( ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 ) -diagnosable, it is equivalent to prove that there is an edge u v E ( A n , k ) with u V ( A n , k ) \ ( F 1 F 2 ) and v F 1 F 2 for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of V ( A n , k ) with | F 1 | ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 and | F 2 | ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 .
We prove this statement by contradiction. Suppose that there are two distinct g-good-neighbor faulty subsets F 1 and F 2 of A n , k with | F 1 | ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 and | F 2 | ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 , but the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition in Theorem 1, i.e., there are no edges between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . Without loss of generality, suppose that F 2 \ F 1 .
Case 1. V ( A n , k ) = F 1 F 2 .
Note that ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 = g 2 2 + ( k 3 ) g + k 1 2 . Since k [ 2 , n 2 ] , g 2 2 + ( k 3 ) g + k 1 2 g 2 2 + ( n 5 ) g + n 2 1 2 . Let y = g 2 2 + ( n 5 ) g + n 2 1 2 . Then, y m a x = 1 2 n 2 4 n + 10 for g = n 5 and g 2 2 + ( k 3 ) g + k 1 2 1 2 n 2 4 n + 10 .
Assume V ( A n , k ) = F 1 F 2 . We have that n ! ( n k ) ! = | V ( A n , k ) | = | F 1 F 2 | = | F 1 | + | F 2 | | F 1 F 2 | | F 1 | + | F 2 | 2 ( ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 ) 2 ( ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 ) 2 ( ( 1 2 n 2 4 n + 10 ) ( n k ) + g + 1 ) = ( n 2 8 n + 20 ) ( n 2 ) + 2 ( n 2 ) + 2 = n 3 10 n 2 + 34 n 34 . When k = 3 , n ! ( n k ) ! = n 3 3 n 2 + 2 n . Note n 3 3 n 2 + 2 n n ! ( n k ) ! for k 3 . Thus, n 3 3 n 2 + 2 n n 3 10 n 2 + 36 n 40 . In fact, n 3 3 n 2 + 2 n > n 3 10 n 2 + 36 n 40 when n 4 . This is a contradiction. Therefore, V ( A n , k ) F 1 F 2 .
Case 2. V ( A n , k ) F 1 F 2
According to the hypothesis, there are no edges between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . Since F 1 is a g-good-neighbor faulty set and A n , k F 1 has two parts A n , k F 1 F 2 and A n , k [ F 2 \ F 1 ] , we have that δ ( A n , k F 1 F 2 ) g and δ ( A n , k [ F 2 \ F 1 ] ) g . Similarly, δ ( A n , k [ F 1 \ F 2 ] ) g when F 1 \ F 2 . Therefore, F 1 F 2 is also a g-good-neighbor faulty set. Since there are no edges between V ( A n , k F 1 F 2 ) and F 1 F 2 , F 1 F 2 is also a g-good-neighbor cut. When F 1 \ F 2 = , F 1 F 2 = F 1 is also a g-good-neighbor faulty set. Since there are no edges between V ( A n , k F 1 F 2 ) and F 1 F 2 , F 1 F 2 is a g-good-neighbor cut. By Lemma 13, | F 1 F 2 | ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + 1 . Since δ ( A n , k [ F 2 \ F 1 ] ) g , | F 2 \ F 1 | g + 1 . Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 F 2 | g + 1 + ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + 1 = ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 2 , which contradicts with that | F 2 | ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 . Thus, A n , k is g-good-neighbor ( ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 ) -diagnosable. By the definition of t g ( A n , k ) , t g ( A n , k ) ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 . The proof is complete. □
Combining Lemmas 15 and 16, we have the following theorem.
Theorem 2.
Let n , k , g be positive integers such that n 4 , 3 k n 2 , 3 g < n k . Then, ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 t g ( A n , k ) [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) under the PMC model.
Theorem 3.
Let A n , k be the arrangement graph with n > k 2 . Then, the diagnosability t ( A n , k ) = k ( n k ) under the PMC model.
Proof. 
Let u V ( A n , k ) . Then, N ( u ) is a cut of A n , k and | N ( u ) | = k ( n k ) . Let F 1 = N ( u ) , F 2 = { u } N ( u ) . Then, | F 1 | = k ( n k ) , | F 2 | = | X | + | F 1 | = k ( n k ) + 1 , δ ( A n , k F 1 ) 0 and δ ( A n , k F 2 ) 0 . Therefore, F 1 and F 2 are 0-good-neighbor faulty sets of A n , k with | F 1 | = k ( n k ) and | F 2 | = k ( n k ) + 1 . We will prove A n , k is not 0-good-neighbor ( k ( n k ) + 1 ) -diagnosable. Since { u } = F 1 F 2 and N ( u ) = F 1 F 2 , there is no edge of A n , k between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . By Theorem 1, we can show that A n , k is not 0-good-neighbor ( k ( n k ) + 1 ) -diagnosable under the PMC model. Hence, by the definition of the 0-good-neighbor diagnosability, we conclude that the 0-good-neighbor diagnosability of A n , k is less than k ( n k ) + 1 , i.e., t 0 ( A n , k ) k ( n k ) .
By Theorem 1, to prove A n , k is 0-good-neighbor k ( n k ) -diagnosable, it is equivalent to prove that there is an edge u v E ( A n , k ) with u V ( A n , k ) \ ( F 1 F 2 ) and v F 1 F 2 for each distinct pair of 0-good-neighbor faulty subsets F 1 and F 2 of V ( A n , k ) with | F 1 | k ( n k ) and | F 2 | k ( n k ) .
We prove this statement by contradiction. Suppose that there are two distinct 0-good-neighbor faulty subsets F 1 and F 2 of A n , k with | F 1 | k ( n k ) and | F 2 | k ( n k ) , but the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition in Theorem 1, i.e., there are no edges between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . Without loss of generality, suppose that F 2 \ F 1 .
Assume V ( A n , k ) = F 1 F 2 . We have that n ! ( n k ) ! = | V ( A n , k ) | = | F 1 F 2 | = | F 1 | + | F 2 | | F 1 F 2 | | F 1 | + | F 2 | 2 k ( n k ) . When k = 2 , n 2 n = n ! ( n 2 ) ! 4 n 8 , a contradiction. Therefore, V ( A n , 2 ) F 1 F 2 . When k = 3 , n ! ( n k ) ! = n 3 3 n 2 + 2 n . Note n 3 3 n 2 + 2 n n ! ( n k ) ! and 2 k ( n k ) 2 n 2 8 n + 6 for k 3 . Thus, n 3 3 n 2 + 2 n 2 n 2 8 n + 6 . In fact, n 3 3 n 2 + 2 n > 2 n 2 8 n + 6 when n 4 . This is a contradiction. Therefore, V ( A n , k ) F 1 F 2 .
According to the hypothesis, there are no edges between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . Since F 1 is a 0-good-neighbor faulty set and A n , k F 1 has two parts A n , k F 1 F 2 and A n , k [ F 2 \ F 1 ] , we have that δ ( A n , k F 1 F 2 ) 0 and δ ( A n , k [ F 2 \ F 1 ] ) 0 . Similarly, δ ( A n , k [ F 1 \ F 2 ] ) 0 when F 1 \ F 2 . Therefore, F 1 F 2 is also a 0-good-neighbor faulty set. Since there are no edges between V ( A n , k F 1 F 2 ) and F 1 F 2 , F 1 F 2 is also a 0-good-neighbor cut. When F 1 \ F 2 = , F 1 F 2 = F 1 is also a 0-good-neighbor faulty set. Since there are no edges between V ( A n , k F 1 F 2 ) and F 1 F 2 , F 1 F 2 is a 0-good-neighbor cut. By Lemma 2, | F 1 F 2 | k ( n k ) . Since δ ( A n , k [ F 2 \ F 1 ] ) 0 , | F 2 \ F 1 | 1 . Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 F 2 | 1 + k ( n k ) , which contradicts with that | F 2 | k ( n k ) . Thus, A n , k is 0-good-neighbor k ( n k ) -diagnosable. By the definition of t 0 ( A n , k ) , t 0 ( A n , k ) k ( n k ) . Therefore, t 0 ( G ) = t ( G ) = k ( n k ) . □
Lemma 17.
Let n 5 and 2 k < n . Then, t 1 ( A n , k ) ( 2 k 1 ) ( n k ) under the PMC model.
Proof. 
By Theorem 1, to prove A n , k is 1-good-neighbor ( 2 k 1 ) ( n k ) -diagnosable, it is equivalent to prove that there is an edge u v E ( A n , k ) with u V ( A n , k ) \ ( F 1 F 2 ) and v F 1 F 2 for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of V ( A n , k ) with | F 1 | ( 2 k 1 ) ( n k ) and | F 2 | ( 2 k 1 ) ( n k ) .
We prove this statement by contradiction. Suppose that there are two distinct 1-good-neighbor faulty subsets F 1 and F 2 of A n , k with | F 1 | ( 3 k 2 ) ( n k ) and | F 2 | ( 3 k 2 ) ( n k ) , but the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition in Theorem 1, i.e., there are no edges between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . Without loss of generality, assume that F 2 \ F 1 .
Assume V ( A n , k ) = F 1 F 2 . We have that n ! ( n k ) ! = | V ( A n , k ) | = | F 1 F 2 | = | F 1 | + | F 2 | | F 1 F 2 | | F 1 | + | F 2 | 2 ( 2 k 1 ) ( n k ) 2 ( 2 n 3 ) ( n 2 ) = 4 n 2 14 n + 12 . When k = 3 , n ! ( n k ) ! = n 3 3 n 2 + 2 n . Note n 3 3 n 2 + 2 n n ! ( n k ) ! for k 3 . Thus, n 3 3 n 2 + 2 n 4 n 2 14 n + 12 . In fact, n 3 3 n 2 + 2 n > 4 n 2 14 n + 12 when n 5 . This is a contradiction. When k = 2 , n 2 n = n ! ( n k ) ! 2 ( 2 k 1 ) ( n k ) = 6 ( n 2 ) . In fact, n 2 n > 6 ( n 2 ) when n 5 . This is a contradiction. Therefore, V ( A n , k ) F 1 F 2 .
According to the hypothesis, there are no edges between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . Since F 1 is a 1-good-neighbor faulty set and A n , k F 1 has two parts A n , k F 1 F 2 and A n , k [ F 2 \ F 1 ] , we have that δ ( A n , k F 1 F 2 ) 1 and δ ( A n , k [ F 2 \ F 1 ] ) 1 . Similarly, δ ( A n , k [ F 1 \ F 2 ] ) 1 when F 1 \ F 2 . Therefore, F 1 F 2 is also a 1-good-neighbor faulty set. Since there are no edges between V ( A n , k F 1 F 2 ) and F 1 F 2 , F 1 F 2 is also a 1-good-neighbor cut. When F 1 \ F 2 = , F 1 F 2 = F 1 is also a 1-good-neighbor faulty set. Since there are no edges between V ( A n , k F 1 F 2 ) and F 1 F 2 , F 1 F 2 is a 1-good-neighbor cut. By Lemma 10, | F 1 F 2 | ( 2 k 1 ) ( n k ) 1 . Since δ ( A n , k [ F 2 \ F 1 ] ) 1 , | F 2 \ F 1 | 2 . Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 F 2 | 2 + ( 2 k 1 ) ( n k ) 1 = ( 2 k 1 ) ( n k ) + 1 , which contradicts with that | F 2 | ( 2 k 1 ) ( n k ) . Thus, A n , k is 1-good-neighbor ( 2 k 1 ) ( n k ) -diagnosable. By the definition of t 1 ( A n , k ) , t 1 ( A n , k ) ( 2 k 1 ) ( n k ) . □
Combining Lemmas 15 and 17, we have the following theorem.
Theorem 4.
Let n 5 and 2 k < n . Then, t 1 ( A n , k ) = ( 2 k 1 ) ( n k ) under the PMC model.
Lemma 18.
Let n 8 and k { i : i = 3 , , n 5 } { n 2 , n 1 } . Then, t 2 ( A n , k ) ( 3 k 2 ) ( n k ) under the PMC model.
Proof. 
By Theorem 1, to prove A n , k is 2-good-neighbor ( 3 k 2 ) ( n k ) -diagnosable, it is equivalent to prove that there is an edge u v E ( A n , k ) with u V ( A n , k ) \ ( F 1 F 2 ) and v F 1 F 2 for each distinct pair of g-good-neighbor faulty subsets F 1 and F 2 of V ( A n , k ) with | F 1 | ( 3 k 2 ) ( n k ) and | F 2 | ( 3 k 2 ) ( n k ) .
We prove this statement by contradiction. Suppose that there are two distinct g-good-neighbor faulty subsets F 1 and F 2 of A n , k with | F 1 | ( 3 k 2 ) ( n k ) and | F 2 | ( 3 k 2 ) ( n k ) , but the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition in Theorem 1, i.e., there are no edges between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . Without loss of generality, assume that F 2 \ F 1 .
Assume V ( A n , k ) = F 1 F 2 . We have that n ! ( n k ) ! = | V ( A n , k ) | = | F 1 F 2 | = | F 1 | + | F 2 | | F 1 F 2 | | F 1 | + | F 2 | 2 ( 3 k 2 ) ( n k ) 2 ( 3 n 5 ) ( n 3 ) = 6 n 2 28 n + 30 . When k = 3 , n ! ( n k ) ! = n 3 3 n 2 + 2 n . Note n 3 3 n 2 + 2 n n ! ( n k ) ! for k 3 . Thus, n 3 3 n 2 + 2 n 6 n 2 28 n + 30 . In fact, n 3 3 n 2 + 2 n > 6 n 2 28 n + 30 when n 8 . This is a contradiction. Therefore, V ( A n , k ) F 1 F 2 .
According to the hypothesis, there are no edges between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . Since F 1 is a 2-good-neighbor faulty set and A n , k F 1 has two parts A n , k F 1 F 2 and A n , k [ F 2 \ F 1 ] , we have that δ ( A n , k F 1 F 2 ) 2 and δ ( A n , k [ F 2 \ F 1 ] ) 2 . Similarly, δ ( A n , k [ F 1 \ F 2 ] ) 2 when F 1 \ F 2 . Therefore, F 1 F 2 is also a 2-good-neighbor faulty set. Since there are no edges between V ( A n , k F 1 F 2 ) and F 1 F 2 , F 1 F 2 is also a 2-good-neighbor cut. When F 1 \ F 2 = , F 1 F 2 = F 1 is also a 2-good-neighbor faulty set. Since there are no edges between V ( A n , k F 1 F 2 ) and F 1 F 2 , F 1 F 2 is a 2-good-neighbor cut. By Lemma 11, | F 1 F 2 | ( 3 k 2 ) ( n k ) 2 . Since δ ( A n , k [ F 2 \ F 1 ] ) 2 , | F 2 \ F 1 | 3 . Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 F 2 | 3 + ( 3 k 2 ) ( n k ) 2 = ( 3 k 2 ) ( n k ) + 1 , which contradicts with that | F 2 | ( 3 k 2 ) ( n k ) . Thus, A n , k is 2-good-neighbor ( 3 k 2 ) ( n k ) -diagnosable. By the definition of t 2 ( A n , k ) , t 2 ( A n , k ) ( 3 k 2 ) ( n k ) . □
Combining Lemmas 15 and 18, we have the following theorem.
Theorem 5.
Let n 8 and k { i : i = 3 , , n 5 } { n 2 , n 1 } . Then, t 2 ( A n , k ) = ( 3 k 2 ) ( n k ) under the PMC model.
For n 8 , A n , 2 is decomposed into n subgraphs A n , 2 1 , , A n , 2 n . By Proposition 1, A n , 2 i is isomorphic to K n 1 for i = 1 , 2 , , n . Let a = ( 1 , n ) , b = ( 2 , n ) , c = ( 1 , n 1 ) , d = ( 2 , n 1 ) . Then, a , b V ( A n , 2 n ) , a b E ( A n , 2 n ) , c , d V ( A n , 2 n 1 ) , c d E ( A n , 2 n 1 ) , a c E ( A n , 2 ) and b d E ( A n , 2 ) , and a b d c a is a 4-cycle of A n , 2 .
Lemma 19.
For n 8 and A n , 2 , let X = { a , b , c , d } be defined as above, and let F 1 = N A n , 2 ( X ) , F 2 = X N A n , 2 ( X ) . Then, | F 1 | = 4 n 12 , | F 2 | = 4 n 8 , δ ( A n , 2 [ X ] ) = 2 and δ ( A n , 2 F 1 F 2 ) 2 .
Proof. 
Note that | N ( X ) | = 4 ( n 3 ) = 4 n 12 . Then, | F 2 | = 4 n 8 . Since a b d c a is a four-cycle of A n , 2 , δ ( A n , 2 [ X ] ) = 2 . Since n 8 , δ ( A n , 2 n { a , b } ) 2 and δ ( A n , 2 n 1 { c , d } ) 2 . Thus, δ ( A n , 2 F 1 F 2 ) 2 . □
Lemma 20.
For n 8 , t 2 ( A n , 2 ) 4 n 9 under the PMC model.
Proof. 
Let X be defined in Lemma 19, and let F 1 = N A n , 2 ( X ) , F 2 = X N A n , 2 ( X ) . By Lemma 19, | F 1 | = 4 n 12 , | F 2 | = | X | + | F 1 | = 4 n 8 , δ ( A n , 2 F 1 ) 2 and δ ( A n , 2 F 2 ) 2 . Therefore, F 1 and F 2 are 2-good-neighbor faulty sets of A n , 2 with | F 1 | = 4 n 12 and | F 2 | = 4 n 8 .
We will prove A n , 2 is not 2-good-neighbor ( 4 n 8 ) -diagnosable. Since X = F 1 F 2 and N A n , k ( X ) = F 1 F 2 , there is no edge of A n , 2 between V ( A n , 2 ) \ ( F 1 F 2 ) and F 1 F 2 . By Theorem 1, we can deduce that A n , 2 is not 2-good-neighbor ( 4 n 8 ) -diagnosable under the PMC model. Hence, by the definition of the 2-good-neighbor diagnosability, we conclude that the 2-good-neighbor diagnosability of A n , 2 is less than 4 n 8 , i.e., t g ( A n , 2 ) 4 n 9 . □
Lemma 21.
For n 8 , t 2 ( A n , 2 ) 4 n 9 under the PMC model.
Proof. 
By Theorem 1, to prove A n , 2 is 2-good-neighbor ( 4 n 9 ) -diagnosable, it is equivalent to prove that there is an edge u v E ( A n , 2 ) with u V ( A n , 2 ) \ ( F 1 F 2 ) and v F 1 F 2 for each distinct pair of 2-good-neighbor faulty subsets F 1 and F 2 of V ( A n , 2 ) with | F 1 | 4 n 9 and | F 2 | 4 n 9 .
We prove this statement by contradiction. Suppose that there are two distinct 2-good-neighbor faulty subsets F 1 and F 2 of A n , 2 with | F 1 | 4 n 9 and | F 2 | 4 n 9 , but the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition in Theorem 1, i.e., there are no edges between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . Without loss of generality, assume that F 2 \ F 1 .
Assume V ( A n , 2 ) = F 1 F 2 . We have that n 2 n = n ! ( n 2 ) ! = | V ( A n , 2 ) | = | F 1 F 2 | = | F 1 | + | F 2 | | F 1 F 2 | | F 1 | + | F 2 | 2 ( 4 n 9 ) = 8 n 18 , a contradiction to n 8 . Therefore, V ( A n , k ) F 1 F 2 .
According to the hypothesis, there are no edges between V ( A n , 2 ) \ ( F 1 F 2 ) and F 1 F 2 . Since F 1 is a 2-good-neighbor faulty set and A n , 2 F 1 has two parts A n , k F 1 F 2 and A n , 2 [ F 2 \ F 1 ] , we have that δ ( A n , 2 F 1 F 2 ) 2 and δ ( A n , 2 [ F 2 \ F 1 ] ) 2 . Similarly, δ ( A n , 2 [ F 1 \ F 2 ] ) 2 when F 1 \ F 2 . Therefore, F 1 F 2 is also a 2-good-neighbor faulty set. Since there are no edges between V ( A n , 2 F 1 F 2 ) and F 1 F 2 , F 1 F 2 is also a 2-good-neighbor cut. When F 1 \ F 2 = , F 1 F 2 = F 1 is also a 2-good-neighbor faulty set. Since there are no edges between V ( A n , 2 F 1 F 2 ) and F 1 F 2 , F 1 F 2 is a 2-good-neighbor cut. By Lemma 11, | F 1 F 2 | 4 n 12 . If | F 1 F 2 | = 4 n 12 , then, by Lemma 12, | F 2 \ F 1 | = 4 . If | F 1 F 2 | = 4 n 11 or 4 n 10 , then | F 2 \ F 1 | 2 , a contradiction to that δ ( A n , 2 [ F 1 \ F 2 ] ) 2 . Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 F 2 | 4 + ( 4 n 12 ) = 4 n 8 , which contradicts with that | F 2 | 4 n 9 . Thus, A n , k is 2-good-neighbor ( 4 n 9 ) -diagnosable. By the definition of t 2 ( A n , k ) , t 2 ( A n , 2 ) 4 n 9 . □
Combining Lemmas 20 and 21, we have the following theorem.
Theorem 6.
Let n 8 . Then, t 2 ( A n , 2 ) = 4 n 9 under the PMC model.

4. The g-Good-Neighbor Diagnosability of Arrangement Graphs under the MM* Model

Before discussing the g-good-neighbor diagnosability of the arrangement graph A n , k under the MM * model (Figure 3), we first give an existing result.
Theorem 7
([1,23]). A system G = ( V , E ) is g-good-neighbor t-diagnosable under the M M * 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 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 .
Lemma 22.
Let n 3 , 2 k < n and 0 g < n k . Then, the g-good-neighbor diagnosability of the arrangement graph A n , k under the MM* model is less than or equal to [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) , i.e., t g ( A n , k ) [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) .
Proof. 
Let X be defined in Lemma 15, and let F 1 = N A n , k ( X ) , F 2 = X N A n , k ( X ) . By Lemma 14, | F 1 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) g , | F 2 | = | X | + | F 1 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 , δ ( A n , k F 1 ) g and δ ( A n , k F 2 ) g . Therefore, F 1 and F 2 are g-good-neighbor faulty sets of A n , k with | F 1 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) g and | F 2 | = [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 .
We will prove that A n , k is not g-good-neighbor ( [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 ) -diagnosable. Since X = F 1 F 2 and N A n , k ( X ) = F 1 F 2 , there is no edge of A n , k between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . By Theorem 7, we can show that A n , k is not g-good-neighbor ( [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 ) -diagnosable under the MM* model. Hence, by the definition of the g-good-neighbor diagnosability, we show that the g-good-neighbor diagnosability of A n , k is less than [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) + 1 , i.e., t g ( A n , k ) [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) . □
Lemma 23.
Let n , k , g be positive integers such that n 4 , 3 k n 2 , 3 g < n k . Then, the arrangement graph A n , k is g-good-neighbor ( ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 ) -diagnosable under the MM* model.
Proof. 
By the definition of the g-good-neighbor diagnosability, it is sufficient to show that A n , k is g-good-neighbor ( ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 ) -diagnosable for 3 g n k 1 .
By Theorem 7, suppose, on the contrary, that there are two distinct g-good-neighbor faulty subsets F 1 and F 2 of A n , k with | F 1 | ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 and | F 2 | ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 , but the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 7. Without loss of generality, assume that F 2 \ F 1 . Similar to the discussion on V ( A n , k ) = F 1 F 2 in Lemma 16, we can show V ( A n , k ) F 1 F 2 .
Claim 1. A n , k F 1 F 2 has no isolated vertex.
Since F 1 is a g-good neighbor faulty set, for an arbitrary vertex u V ( A n , k ) \ F 1 , | N A n , k F 1 ( u ) | g . Suppose, on the contrary, that A n , k F 1 F 2 has at least one isolated vertex x. Since F 1 is a g-good neighbor faulty set and g 3 , 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 condition in Theorem 7, by Condition (3) of Theorem 7, a contradiction. Therefore, there are at most one vertex u F 2 \ F 1 such that u are adjacent to x. Thus, | N A n , k F 1 ( x ) | = 1 , a contradiction to that F 1 is a g-good neighbor faulty set, where g 3 . Thus, A n , k F 1 F 2 has no isolated vertex. The proof of Claim 1 is complete.
Let u V ( A n , k ) \ ( F 1 F 2 ) . By Claim 1, δ ( A n , k F 1 F 2 ) 1 . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 7, by the condition (1) of Theorem 7, for any pair of adjacent vertices u , w V ( A n , k ) \ ( F 1 F 2 ) , there is no vertex v F 1 F 2 such that u w E ( A n , k ) and u v E ( A n , k ) . It follows that u has no neighbor in F 1 F 2 . Since u is taken arbitrarily, there is no edge between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 .
Since F 2 \ F 1 and F 1 is a g-good-neighbor faulty set, we have that δ A n , k ( [ F 2 \ F 1 ] ) g , δ ( A n , k F 2 F 1 ) g and | F 2 \ F 1 | g + 1 . Since both F 1 and F 2 are g-good-neighbor faulty sets, and there is no edge between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 , F 1 F 2 is a g-good-neighbor cut of A n , k . By Lemma 13, we have | F 1 F 2 | ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + 1 . Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 F 2 | g + 1 + ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + 1 = ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 2 , which contradicts | F 2 | ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 . Therefore, A n , k is g-good-neighbor ( ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 ) -diagnosable and t g ( A n , k ) ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 . The proof is complete. □
Combining Lemmas 22 and 23, we have the following theorem.
Theorem 8.
Let n , k , g be positive integers such that n 4 , 3 k n 2 , 3 g < n k . Then, ( ( g + 1 ) ( k 2 ) + 2 ( g + 1 ) 2 2 ) ( n k ) + g + 1 t g ( A n , k ) [ ( g + 1 ) ( k 1 ) + 1 ] ( n k ) under the MM* model.
Theorem 9
([34]). Let A n , k be an n-dimensional arrangement graph and 3 k < n . Then, the diagnosability of A n , k is k ( n k ) , i.e., t ( A n , k ) = k ( n k ) under the MM* model.
Lemma 24
([30]). A n , k is hamiltonian for 1 k n 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.
Theorem 10
([33]). A graph G = ( V , E ) has a perfect matching if and only if o ( G S ) | S | for all S V .
Lemma 25.
Let n 8 and 2 k < n . Then, t 1 ( A n , k ) ( 2 k 1 ) ( n k ) under the MM * model.
Proof. 
By the definition of 1-good-neighbor diagnosability, it is sufficient to show that A n , k is 1-good-neighbor ( 2 k 1 ) ( n k ) -diagnosable.
By Theorem 7, suppose, on the contrary, that there are two distinct 1-good-neighbor faulty subsets F 1 and F 2 of A n , k with | F 1 | ( 2 k 1 ) ( n k ) and | F 2 | ( 2 k 1 ) ( n k ) , but the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 7. Without loss of generality, suppose that F 2 \ F 1 . Assume V ( A n , k ) = F 1 F 2 . We have that n ! ( n k ) ! = | V ( A n , k ) | = | F 1 F 2 | = | F 1 | + | F 2 | | F 1 F 2 | | F 1 | + | F 2 | 2 ( 2 k 1 ) ( n k ) . When k = 2 , n 2 n = n ! ( n 2 ) ! = | V ( A n , 2 ) | = | F 1 F 2 | 6 n 12 , a contradiction to n 5 . Therefore, V ( A n , 2 ) F 1 F 2 . When k = 3 , n ! ( n k ) ! = n 3 3 n 2 + 2 n . Note n 3 3 n 2 + 2 n n ! ( n k ) ! for k 3 . Thus, 2 ( 2 k 1 ) ( n k ) 2 ( 2 ( n 1 ) 1 ) ( n 3 ) 4 n 2 18 n + 18 . In fact, n 3 3 n 2 + 2 n > 4 n 2 18 n + 18 when n 5 . This is a contradiction. Therefore, V ( A n , k ) F 1 F 2 .
Claim 1. A n , k F 1 F 2 has no isolated vertex.
Suppose, on the contrary, that A n , k 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 condition in Theorem 7, 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. Similarly, we can show that there is just a vertex v F 1 \ F 2 such that v is adjacent to w when F 1 \ F 2 . Suppose F 1 \ F 2 = . Then, F 1 F 2 . Since F 2 is a 1-good neighbor faulty set, A n , k F 2 = A n , k F 1 F 2 has no isolated vertex. Therefore, F 1 \ F 2 as follows. Let W V ( A n , k ) \ ( F 1 F 2 ) be the set of isolated vertices in A n , k [ V ( A n , k ) \ ( F 1 F 2 ) ] , and let H be the subgraph induced by the vertex set V ( A n , k ) \ ( F 1 F 2 W ) . Then, for any w W , there are ( k ( n k ) 2 ) neighbors in F 1 F 2 . Since | V ( A n , k ) | is even and Lemma 24, A n , k has a perfect matching. By Theorem 10, | W | o ( G ( F 1 F 2 ) ) | F 1 F 2 | | F 1 | + | F 2 | | F 1 F 2 | 2 ( 2 k 1 ) ( n k ) ( k ( n k ) 2 ) = ( n k ) ( 3 k 2 ) + 2 3 n 2 11 n + 12 . In particular, | W | 4 n 6 when k = 2 . When k = 2 , n 2 n = | V ( A n , 2 ) | = | F 1 F 2 | + | W | 2 ( 4 n 6 ) = 8 n 12 . This is a contradiction to n 8 . Thus, V ( H ) . When k = 3 , n ! ( n k ) ! = n 3 3 n 2 + 2 n . Note n 3 3 n 2 + 2 n n ! ( n k ) ! for k 3 . Note that n 3 3 n 2 + 2 n = | V ( A n , k ) | = | F 1 F 2 | + | W | 2 ( 3 n 2 11 n + 12 ) = 6 n 2 22 n + 24 . This is a contradiction to n 8 . Thus, V ( H ) . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with the condition (1) of Theorem 7, and any vertex of V ( H ) is not isolated in H, we show that there is no edge between V ( H ) and F 1 F 2 . Thus, F 1 F 2 is a vertex cut of A n , k and δ ( A n , k ( F 1 F 2 ) ) 1 , i.e., F 1 F 2 is a 1-good-neighbor cut of A n , k . By Lemma 10, | F 1 F 2 | ( 2 k 1 ) ( n k ) 1 . Because | F 1 | ( 2 k 1 ) ( n k ) , | F 2 | ( 2 k 1 ) ( n k ) , and neither F 1 \ F 2 nor F 2 \ F 1 is empty, we have | F 1 \ F 2 | = | F 2 \ F 1 | = 1 . 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 . Suppose that v 1 is adjacent to v 2 . Then, v 1 v 2 v v 1 is a three-cycle and | N ( { v 1 , v 2 , v } ) | = 3 [ ( k 1 ) ( n k ) 1 ] + n k + 1 > ( 2 k 1 ) ( n k ) 1 | F 1 F 2 | , a contradiction. Therefore, suppose that v 1 is not adjacent to v 2 . According to Lemma 9, there are at most two common neighbors for any pair of vertices in A n , k , it follows that there are at most three isolated vertices in A n , k F 1 F 2 , i.e., | W | 2 .
Suppose that there is exactly one isolated vertex v in A n , k F 1 F 2 . Let v 1 and v 2 be adjacent to v. Then, N A n , k ( v ) \ { v 1 , v 2 } F 1 F 2 and | N A n , k ( v ) ( F 1 F 2 ) | = k ( n k ) 2 . Note that | N A n , k ( v 1 ) ( F 1 F 2 ) | = k ( n k ) 1 and | N A n , k ( v 2 ) ( F 1 F 2 ) | = k ( n k ) 1 . By Lemma 9, | F 1 F 2 | k ( n k ) 2 + k ( n k ) 1 + k ( n k ) 1 2 ( n k 1 ) 2 = ( 3 k 2 ) ( n k ) 4 . It follows that | F 2 | = | F 2 \ F 1 | + | F 1 F 2 | 1 + ( 3 k 2 ) ( n k ) 4 = ( 3 k 2 ) ( n k ) 3 > ( 2 k 1 ) ( n k ) ( n 8 ) , which contradicts | F 2 | ( 2 k 1 ) ( n k ) .
Suppose that there are exactly two isolated vertices v and w in A n , k F 1 F 2 . Let v 1 and v 2 be adjacent to v and w, respectively. Since v 1 , v 2 N A n , k ( { v , w } ) , by Lemma 9, | N A n , k ( { v , w } ) ( F 1 F 2 ) | = 2 ( k ( n k ) 2 ) . Note that | F 1 F 2 | ( 2 k 1 ) ( n k ) 1 . If n > k + 3 , then 2 ( k ( n k ) 2 ) > ( 2 k 1 ) ( n k ) 1 , a contradiction. Thus, n k + 3 . Since n k + 1 , k + 1 n k + 3 . If n = k + 1 , then, by Lemma 9, a contradiction to | W | = 2 . Suppose that n = k + 2 . Then, A n , k = A n , n 2 . By the proof of Lemma 3.2 ([18]), A n , n 2 F 1 F 2 has no isolated vertex. Suppose that n = k + 3 . Then, 2 ( k ( n k ) 2 ) = ( 2 k 1 ) ( n k ) 1 = 6 n 22 . By Lemma 1, let v 1 = ( 1 , 2 , , n 4 , n 3 ) .Without loss of generality, suppose v = ( 1 , 2 , , n 4 , n ) and w = ( n , 2 , , n 4 , n 3 ) . Then, the vertex v = ( 1 , n 1 , , n 4 , n 3 ) is not adjacent to v and w. Thus, | F 1 F 2 | > ( 2 k 1 ) ( n k ) 1 , a contradiction. The proof of Claim 1 is complete.
Let u V ( A n , k ) \ ( F 1 F 2 ) . By Claim 1, u has at least one neighbor in A n , k F 1 F 2 . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 7, by the condition (1) of Theorem 7, for any pair of adjacent vertices u , w V ( A n , k ) \ ( F 1 F 2 ) , there is no vertex v F 1 F 2 such that u w E ( A n , k ) and v w E ( A n , k ) . It follows that u has no neighbor in F 1 F 2 . Since u is taken arbitrarily, there is no edge between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 . Since F 2 \ F 1 and F 1 is a 1-good-neighbor faulty set, δ A n , k ( [ F 2 \ F 1 ] ) 1 and | 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 ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 , F 1 F 2 is a 1-good-neighbor cut of A n , k . By Lemma 10, we have | F 1 F 2 | ( 2 k 1 ) ( n k ) 1 . Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 F 2 | 2 + ( ( 2 k 1 ) ( n k ) 1 ) = ( 2 k 1 ) ( n k ) + 1 , which contradicts | F 2 | ( 3 k 2 ) ( n k ) . Therefore, A n , k is 1-good-neighbor ( 3 k 2 ) ( n k ) -diagnosable and t 1 ( A n , k ) ( 3 k 2 ) ( n k ) . The proof is complete. □
Combining Lemmas 22 and 25, we have the following theorem.
Theorem 11.
Let n 8 . Then, t 1 ( A n , k ) = ( 2 k 1 ) ( n k ) under the MM * model.
Lemma 26.
Let n 8 and k { i : i = 3 , , n 5 } { n 2 , n 1 } . Then, t 2 ( A n , k ) ( 3 k 2 ) ( n k ) under the MM * model.
Proof. 
By the definition of the 2-good-neighbor diagnosability, it is sufficient to show that A n , k is g-good-neighbor ( 3 k 2 ) ( n k ) -diagnosable.
By Theorem 7, suppose, on the contrary, that there are two distinct g-good-neighbor faulty subsets F 1 and F 2 of A n , k with | F 1 | ( 3 k 2 ) ( n k ) and | F 2 | ( 3 k 2 ) ( n k ) , but the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 7. Without loss of generality, suppose that F 2 \ F 1 . Similar to the discussion on V ( A n , k ) = F 1 F 2 in Lemma 18, we have V ( A n , k ) F 1 F 2 .
Claim 1. A n , k F 1 F 2 has no isolated vertex.
Since F 1 is a 2-good neighbor faulty set, for an arbitrary vertex u V ( A n , k ) \ F 1 , | N A n , k F 1 ( u ) | 2 . Suppose, on the contrary, that A n , k F 1 F 2 has at least one isolated vertex x. Since F 1 is a 2-good neighbor faulty set, 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 condition in Theorem 7, by the condition (3) of Theorem 7, a contradiction. Therefore, there are at most one vertex u F 2 \ F 1 such that u are adjacent to x. Thus, | N A n , k F 1 ( x ) | = 1 , a contradiction to that F 1 is a 2-good neighbor faulty set. Thus, A n , k F 1 F 2 has no isolated vertex. The proof of Claim 1 is complete.
Let u V ( A n , k ) \ ( F 1 F 2 ) . By Claim 1, δ ( A n , k F 1 F 2 ) 1 . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 7, by the condition (1) of Theorem 7, for any pair of adjacent vertices u , w V ( A n , k ) \ ( F 1 F 2 ) , there is no vertex v F 1 F 2 such that u w E ( A n , k ) and u v E ( A n , k ) . It follows that u has no neighbor in F 1 F 2 . Since u is taken arbitrarily, there is no edge between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 .
Since F 2 \ F 1 and F 1 is a 2-good-neighbor faulty set, we have that δ A n , k ( [ F 2 \ F 1 ] ) 2 , δ ( A n , k F 2 F 1 ) 2 and | F 2 \ F 1 | 2 + 1 = 3 . Since both F 1 and F 2 are 2-good-neighbor faulty sets, and there is no edge between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 , F 1 F 2 is a 2-good-neighbor cut of A n , k . By Lemma 11, we have | F 1 F 2 | ( 3 k 2 ) ( n k ) 2 . Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 F 2 | 3 + ( 3 k 2 ) ( n k ) 2 = ( 3 k 2 ) ( n k ) + 1 , which contradicts | F 2 | ( 3 k 2 ) ( n k ) . Therefore, A n , k is 2-good-neighbor ( 3 k 2 ) ( n k ) -diagnosable and t 2 ( A n , k ) ( 3 k 2 ) ( n k ) . The proof is complete. □
Combining Lemmas 22 and 26, we have the following theorem.
Theorem 12.
Let n 8 and k { i : i = 3 , , n 5 } { n 2 , n 1 } . Then, t 2 ( A n , k ) = ( 3 k 2 ) ( n k ) under the MM * model.
Lemma 27.
For n 8 , t 2 ( A n , 2 ) 4 n 9 under the MM * model.
Proof. 
Let X be defined in Lemma 19, and let F 1 = N A n , 2 ( X ) , F 2 = X N A n , 2 ( X ) . By Lemma 19, | F 1 | = 4 n 12 , | F 2 | = | X | + | F 1 | = 4 n 8 , δ ( A n , 2 F 1 ) 2 and δ ( A n , 2 F 2 ) 2 . Therefore, F 1 and F 2 are 2-good-neighbor faulty sets of A n , 2 with | F 1 | = 4 n 12 and | F 2 | = 4 n 8 .
We will prove A n , 2 is not 2-good-neighbor ( 4 n 8 ) -diagnosable. Since X = F 1 F 2 and N A n , k ( X ) = F 1 F 2 , there is no edge of A n , 2 between V ( A n , 2 ) \ ( F 1 F 2 ) and F 1 F 2 . By Theorem 7, we show that A n , 2 is not 2-good-neighbor ( 4 n 8 ) -diagnosable under the MM * model. Hence, by the definition of the 2-good-neighbor diagnosability, we show that the 2-good-neighbor diagnosability of A n , 2 is less than 4 n 8 , i.e., t g ( A n , 2 ) 4 n 9 . □
Lemma 28.
For n 8 , t 2 ( A n , 2 ) 4 n 9 under the MM * model.
Proof. 
By the definition of the 2-good-neighbor diagnosability, it is sufficient to show that A n , k is 2-good-neighbor ( 4 n 9 ) -diagnosable.
By Theorem 7, suppose, on the contrary, that there are two distinct 2-good-neighbor faulty subsets F 1 and F 2 of A n , k with | F 1 | 4 n 9 and | F 2 | 4 n 9 , but the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 7. Without loss of generality, suppose that F 2 \ F 1 . Similar to the discussion on V ( A n , k ) = F 1 F 2 in Lemma 21, we have V ( A n , k ) F 1 F 2 .
Claim 1. A n , k F 1 F 2 has no isolated vertex.
Since F 1 is a 2-good neighbor faulty set, for an arbitrary vertex u V ( A n , k ) \ F 1 , | N A n , k F 1 ( u ) | 2 . Suppose, on the contrary, that A n , k F 1 F 2 has at least one isolated vertex x. Since F 1 is a 2-good neighbor faulty set, 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 condition in Theorem 7, by the condition (3) of Theorem 7, a contradiction. Therefore, there are at most one vertex u F 2 \ F 1 such that u are adjacent to x. Thus, | N A n , k F 1 ( x ) | = 1 , a contradiction to that F 1 is a 2-good neighbor faulty set. Thus, A n , k F 1 F 2 has no isolated vertex. The proof of Claim 1 is complete.
Let u V ( A n , k ) \ ( F 1 F 2 ) . By Claim 1, δ ( A n , k F 1 F 2 ) 1 . Since the vertex set pair ( F 1 , F 2 ) is not satisfied with any condition in Theorem 7, by the condition (1) of Theorem 7, for any pair of adjacent vertices u , w V ( A n , k ) \ ( F 1 F 2 ) , there is no vertex v F 1 F 2 such that u w E ( A n , k ) and u v E ( A n , k ) . It follows that u has no neighbor in F 1 F 2 . Since u is taken arbitrarily, there is no edge between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 .
Since F 2 \ F 1 and F 1 is a 2-good-neighbor faulty set, we have that δ A n , k ( [ F 2 \ F 1 ] ) 2 , δ ( A n , k F 2 F 1 ) 2 and | F 2 \ F 1 | 2 + 1 = 3 . Since both F 1 and F 2 are 2-good-neighbor faulty sets, and there is no edge between V ( A n , k ) \ ( F 1 F 2 ) and F 1 F 2 , F 1 F 2 is a 2-good-neighbor cut of A n , k . By Lemma 11, we have | F 1 F 2 | 4 n 12 . If | F 1 F 2 | = 4 n 12 , then, by Lemma 12, | F 2 \ F 1 | = 4 . If | F 1 F 2 | = 4 n 11 or 4 n 10 , then | F 2 \ F 1 | 2 , a contradiction to that δ ( A n , 2 [ F 1 \ F 2 ] ) 2 . Therefore, | F 2 | = | F 2 \ F 1 | + | F 1 F 2 | 4 + ( 4 n 12 ) = 4 n 8 , which contradicts with that | F 2 | 4 n 9 . Therefore, A n , k is 2-good-neighbor ( 4 n 9 ) -diagnosable and t 2 ( A n , k ) 4 n 9 . The proof is complete. □
Combining Lemmas 27 and 28, we have the following theorem.
Theorem 13.
Let n 8 . Then, t 2 ( A n , 2 ) = 4 n 9 under the MM * model.

5. Conclusions

The conditional diagnosability of a multiprocessor system is an important research topic for fault tolerance of the system. In this paper, we investigate the problem of g-good-neighbor diagnosability of the ( n , k ) -arrangement graph A n , k , and present the g-good-neighbor diagnosability of A n , k under the PMC model and MM* model. The work will help engineers to develop more different networks.

Author Contributions

S.W.and Y.R. conceived and designed the study and wrote the manuscript. S.W. revised the manuscript. All authors read and approved the final manuscript.

Funding

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

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. The arrangement graph A 4 , 2 .
Figure 1. The arrangement graph A 4 , 2 .
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Figure 2. Illustration of a distinguishable pair ( F 1 , F 2 ) under the PMC model.
Figure 2. Illustration of a distinguishable pair ( F 1 , F 2 ) under the PMC model.
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Figure 3. Illustration of a distinguishable pair ( F 1 , F 2 ) under the MM* model.
Figure 3. Illustration of a distinguishable pair ( F 1 , F 2 ) under the MM* model.
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