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

: 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.


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)

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 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 uw, vw ∈ E(G).We usually assume that the testing result is reliable (respectively, unreliable) if the node u is fault-free (respectively, faulty 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 uw, vw ∈ 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) 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 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 : 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 . 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, A n,k ) with p i = q i for some i ∈ k and p j = q j for all j ∈ k \ {i}}.

Definition 2. Given two positive integers n and k with
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. ([8]
) A graph is vertex-transitive if and only if for its any pair of 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 its any pair of edges (u, v) and (x, y), there exists an automorphism of the graph that maps (u, v) to (x, y).

Lemma 6. ([4]
) Let n, k, g be positive integers such that n ≥ 4, 2 ≤ k ≤ n − 2, g ≥ 3. Then 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.

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 2 ([26]). κ(A
Lemma 5 ([28]).For n ≥ 8, κ (2) (A n,2 ) = 4n − 12, and, for k ∈ {i 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.

Corollary 1. The edge connectivity
to the jth position.For simplicity, we shall take j as the last position k, and use 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 {(12i), (1i2), 3 ≤ i ≤ n} is a generating set for A n .The n-dimensional alternating group graph AG n is the graph with vertex set V(AG n ) = A n in which two vertices u, v are adjacent if and only if Definition 5.The n-dimensional star graph denoted by S n .The vertex set of S n is {u Lemma 8 ([29]).(1).The arrangement graph A n,n−2 is isomorphic to the n-dimensional alternating group graph AG 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:

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 PMC 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-good-neighbor faulty subsets F 1 and
Lemma 13 ([28]).Let n, k, g be positive integers such that n ≥ 4, 2 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 Proof.Let X be defined as above.By the process of the proof of Lemma 13 in [28], Then, the g-good-neighbor diagnosability of the arrangement graph A n,k under the PMC model is less than or equal to Proof.Let X be defined as above, and let 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 and v ∈ F 1 F 2 for each distinct pair of g-good-neighbor faulty subsets F 1 and We prove this statement by contradiction.Suppose that there are two distinct g-good-neighbor is not satisfied with the condition in Theorem 1, i.e., there are no edges between Note that 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 Therefore, F 1 ∩ F 2 is also a g-good-neighbor faulty set.Since there are no edges between is also a g-good-neighbor faulty set.Since there are no edges between The proof is complete.
Combining Lemmas 15 and 16, we have the following theorem.
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 uv ∈ 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 We prove this statement by contradiction.Suppose that there are two distinct 0-good-neighbor faulty subsets F 1 and 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 According to the hypothesis, there are no edges between also a 0-good-neighbor faulty set.Since there are no edges between also a 0-good-neighbor faulty set.Since there are no edges between Lemma 17.Let n ≥ 5 and 2 ≤ k < n.Then, t 1 (A n,k ) ≥ (2k − 1)(n − k) under the PMC model.

Proof. By Theorem 1, to prove
F 2 for each distinct pair of g-good-neighbor faulty subsets F 1 and We prove this statement by contradiction.Suppose that there are two distinct 1-good-neighbor faulty subsets F 1 and is not satisfied with the condition in Theorem 1, i.e., there are no edges between According to the hypothesis, there are no edges between also a 1-good-neighbor faulty set.Since there are no edges between Combining Lemmas 15 and 17, we have the following theorem.
F 2 for each distinct pair of g-good-neighbor faulty subsets F 1 and We prove this statement by contradiction.Suppose that there are two distinct g-good-neighbor faulty subsets F 1 and is not satisfied with the condition in Theorem 1, i.e., there are no edges between According to the hypothesis, there are no edges between also a 2-good-neighbor faulty set.Since there are no edges between also a 2-good-neighbor faulty set.Since there are no edges between Combining Lemmas 15 and 18, we have the following theorem.Theorem 5. Let n ≥ 8 and k ∈ {i ) and bd ∈ E(A n,2 ), and abdca 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 such that uw ∈ E and vw ∈ E. (3) There are two vertices u, v ∈ F 2 \ F 1 and there is a vertex w ∈ Proof.Let X be defined in Lemma 15, and let -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 Proof.By the definition of the g-good-neighbor diagnosability, it is sufficient to show that A n,k is By Theorem 7, suppose, on the contrary, that there are two distinct g-good-neighbor faulty subsets F 1 and is not satisfied with any condition in Theorem 7. Without loss of generality, assume that 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, 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 uw ∈ E(A n,k ) and uv ∈ 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 Since F 2 \ F 1 = ∅ and F 1 is a g-good-neighbor faulty set, we have that Since both F 1 and F 2 are g-good-neighbor faulty sets, and there is no edge between 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 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 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. Proof.By the definition of 1-good-neighbor diagnosability, it is sufficient to show that A n,k is 1-good-neighbor (2k − 1)(n − k)-diagnosable.
By Theorem 7, suppose, on the contrary, that there are two distinct 1-good-neighbor faulty subsets is not satisfied with any condition in Theorem 7. Without loss of generality, suppose that 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 and let H be the subgraph induced by the vertex set 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, 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 Suppose that there is exactly one isolated vertex 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 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 uw ∈ E(A n,k ) and vw ∈ 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 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 uw ∈ E(A n,k ) and uv ∈ 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 Combining Lemmas 27 and 28, we have the following theorem.

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.
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

Figure 2 .
Figure 2. Illustration of a distinguishable pair (F 1 , F 2 ) under the PMC model.