The g-Good-Neighbor Diagnosability of Bubble-Sort Graphs under Preparata , Metze , and Chien ’ s ( PMC ) Model and Maeng and Malek ’ s ( MM ) ∗ Model

Diagnosability of a multiprocessor system is an important topic of study. A measure for fault diagnosis of the system restrains that every fault-free node has at least g fault-free neighbor vertices, which is called the g-good-neighbor diagnosability of the system. As a famous topology structure of interconnection networks, the n-dimensional bubble-sort graph Bn has many good properties. In this paper, we prove that (1) the 1-good-neighbor diagnosability of Bn is 2n− 3 under Preparata, Metze, and Chien’s (PMC) model for n ≥ 4 and Maeng and Malek’s (MM)∗ model for n ≥ 5; (2) the 2-good-neighbor diagnosability of Bn is 4n− 9 under the PMC model and the MM∗ model for n ≥ 4; (3) the 3-good-neighbor diagnosability of Bn is 8n− 25 under the PMC model and the MM∗ model for n ≥ 7.


Introduction
A multiprocessor system and interconnection network (networks for short) have an underlying topology, which is usually presented by a graph, where nodes represent processors and links represent communication links between processors.We use graphs and networks interchangeably.For the system, some processors may fail in the system, so processor fault identification plays an important role in 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 is said to be t-diagnosable if all faulty processors can be identified without replacement, provided that the number of faulty processors presented does not exceed t.The diagnosability t(G) of a system G is the maximum value of t such that G is t-diagnosable.Several diagnosis models (e.g., Preparata, Metze, and Chien's (PMC) model [1], Barsi, Grandoni, and Maestrini's (BGM) model [2], and Maeng and Malek's (MM) model [3]) have been proposed to investigate the diagnosability of multiprocessor systems.In particular, two of the proposed models, the PMC model and MM model, are well known and widely used.In the PMC model, the diagnosis of the system is achieved through two linked processors testing each other.In the MM model, to diagnose a system, a node sends the same task to two of its neighbor vertices, and then compares their responses.Sengupta and Dahbura [4] proposed a special case of the MM model, called the MM* model, in which each node must test all the pairs of its adjacent nodes.In 2012, Peng et al. [5] proposed a measure for fault diagnosis of the system, namely, the g-good-neighbor diagnosability of the system (which is also called g-good-neighbor conditional diagnosability), which requires that every fault-free node contains at least g fault-free neighbors.In [5], they studied the g-good-neighbor diagnosability of the n-dimensional hypercube under the PMC model.Numerous studies have been investigated under the PMC model and MM model or MM* model, see .
In this paper, we prove that (1) the diagnosability of n-dimensional bubble-sort graph B n is n − 1 under the PMC model for n ≥ 4; (2) the 1-good-neighbor diagnosability of B n is 2n − 3 under the PMC model for n ≥ 4 and the MM * model for n ≥ 5; (3) the 2-good-neighbor diagnosability of B n is 4n − 9 under the PMC model and the MM * model for n ≥ 4; (4) the 3-good-neighbor diagnosability of B n is 8n − 25 under the PMC model and the MM * model for n ≥ 7.

Preliminaries
In this section, some definitions and notations needed are introduced for our discussion, then bubble-sort graphs will be introduced.

Definitions and Notations
A multiprocessor system is modeled as an undirected simple graph G = (V, E), whose vertices (nodes) represent processors and edges (links) represent communication links.Given a nonempty vertex subset V of V, the induced subgraph by V in G, denoted by G[V ], is a graph, whose vertex set is V and the edge set is the set of all the edges of G with both endpoints in V .The degree d G (v) of a vertex v is the number of edges incident with v.We denote by δ(G) the minimum degrees of vertices of G.For any vertex v, we define the neighborhood N G (v) of v in G to be the set of vertices adjacent to v. u is called a neighbor vertex or a neighbor of v for u ∈ N G (v).Let S ⊆ V. We use N G (S) to denote the set ∪ v∈S N G (v)\S.For neighborhoods and degrees, we will usually omit the subscript for the graph when no confusion arises.A graph G is said to be k-regular if for any vertex v, d G (v) = k.A graph is bipartite if its vertex set can be partitioned into two subsets X and Y so that every edge has one end in X and one end in Y ; such a partition (X, Y) is called a bipartition of the graph, and X and Y its parts.We denote a bipartite graph G with bipartition (X, Y) by G = (X, Y; E).If G = (X, Y; E) is simple and every vertex in X is joined to every vertex in Y, then G = (X, Y; E) is called a complete bipartite graph, denoted by K n,m , where |X| = n and |Y| = m.Let G = (V, E) be a connected graph.The connectivity κ(G) of a graph G is the minimum number of vertices whose removal results in a disconnected graph or only one vertex left.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 a graph 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).For graph-theoretical terminology and notation not defined here we follow [22].

The Bubble-Sort Graph
The bubble-sort graph has been known as a famous topology structure of interconnection networks.In this section, its definition and some useful properties are introduced.

The Diagnosability of the Bubble-Sort Graph under the PMC Model
In this section, we shall show the g-good-neighbor diagnosability of the bubble-sort graph under the PMC model for g = 0, 1, 2, 3.
Let F 1 and F 2 be two distinct subsets of V for a system G = (V, E).Define the symmetric difference [20] presented a sufficient and necessary condition for a system to be g-good-neighbor t-diagnosable under the PMC model.

Lemma 1 ([20]).
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 F 2 of V with |F 1 | ≤ t and |F 2 | ≤ t (See Figure 2).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.
We prove this statement by contradiction.

110
Suppose that there are two distinct faulty subsets F 1 and but the vertex set pair (F 1 , F 2 ) is not satisfied with the condition in Theorem 1, i.e., there are no edges Without loss of generality, assume that Since there are no edges between 2)-diagnosable under the PMC model.Hence, by the definition of the 1-good-neighbor diagnosability, 129 By the definition of the 1-good-neighbor diagnosability, it is sufficient to show that B n is 132 and v ∈ F 1 F 2 for each distinct pair of 1-good-neighbor faulty subsets F 1 and

134
We prove this statement by contradiction.Suppose that there are two distinct 1-good-neighbor 136 is not satisfied with the condition in Lemma 1, i.e., there are no edges between Proof.
By Lemma 1, we show that B n is not n-diagnosable under the PMC model.Hence, by the definition of the diagnosability, we have that the diagnosability of B n is less than By the definition of the diagnosability, it is sufficient to show that B n is (n We prove this statement by contradiction.Suppose that there are two distinct faulty subsets F 1 and is not satisfied with the condition in Theorem 1, i.e., there are no edges between V(B n )\(F 1 ∪ F 2 ) and F 1 F 2 .Without loss of generality, assume that Theorem 7. The 1-good-neighbor diagnosability of B n is 2n − 3 under the PMC model when n ≥ 4.
By Cases 1 and 2, and F 1 is a 2-good-neighbor cut of B n .When n = 4, it is easy to verify that F 1 is a 2-good-neighbor cut of B n .Lemma 3. Let n ≥ 4. Then the 2-good-neighbor diagnosability t 2 (B n ) ≤ 4n − 9 under the PMC model.

Proof.
Let A be defined in Lemma 6, and let By Lemma 1, we can deduce that B n is not 3-good-neighbor (8n − 24)-diagnosable under the PMC model.Hence, by the definition of 3-good-neighbor diagnosability, we conclude that the 2-good-neighbor diagnosability of B n is less than 8n − 24, i.e., t 2 (B n ) ≤ 8n − 25.Proof.By the definition of 3-good-neighbor diagnosability, it is sufficient to show that B n is 3-good-neighbor (8n − 25)-diagnosable.By Lemma 1, to prove B n is 3-good-neighbor (8n − 25)-diagnosable, it is equivalent to prove that there is an edge uv F 2 for each distinct pair of 3-good-neighbor faulty subsets F 1 and We prove this statement by contradiction.Suppose that there are two distinct 3-good-neighbor faulty subsets F 1 and F 2 of V(B n ) with |F 1 | ≤ 8n − 25 and |F 2 | ≤ 8n − 25, but the vertex set pair (F 1 , F 2 ) is not satisfied with the condition in Lemma 1, i.e., there are no edges between V(B n )\(F 1 ∪ F 2 ) and F 1 F 2 .Without loss of generality, assume that Since there are no edges between also a 3-good-neighbor faulty set.Since there are no edges between Combining Lemmas 7 and 9, we have the following theorem.Theorem 9. Let n ≥ 7. Then the 3-good-neighbor diagnosability of the bubble-sort graph B n under the PMC model is 8n − 25.

The Diagnosability of the Bubble-Sort Graph B n under the MM * Model
Before discussing the diagnosability of the bubble-sort graph B n under the MM * model, we first give an existing result.
Lemma 10 ([4,20]).A system G = (V, E) is g-good-neighbor t-diagnosable under the MM * 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 uw ∈ E and vw ∈ 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 uw ∈ E and vw ∈ 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 uw ∈ E and vw ∈ E (See Figure 3).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.
Version December 22, 2018 submitted to Information 9 of 13 such that uw ∈ E and vw ∈ E (See Fig. 3).The g-good-neighbor diagnosability t g (G) of G is the maximum 319 value of t such that G is g-good-neighbor t-diagnosable under the MM * model.
. By Proposition
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.
Since the bubble-sort graph is a regular bipartite graph, we have the following corollary by Lemma 12.
Corollary 1.The bubble-sort graph has a perfect matching.Lemma 13.Let n ≥ 4. Then the 1-good-neighbor diagnosability of the bubble-sort graph B n under the MM * model is less than or equal to 2n − 3, i.e., t 1 (B n ) ≤ 2n − 3.
Proof.Let u = (1) and v = (12).Then u is adjacent to v. Let F 1 = N({u, v}) and By Lemma 10, we show that B n is not 1-good-neighbor (2n − 2)-diagnosable under the MM * model.Hence, by the definition of the 1-good-neighbor diagnosability, we have that t 1 (B n ) ≤ 2n − 3. Lemma 14.Let n ≥ 5. Then the 1-good-neighbor diagnosability of the bubble-sort graph B n under the MM * model is more than or equal to 2n − 3, i.e., t Since both F 1 and F 2 are 1-good-neighbor faulty sets, and there is no edge between Combining Lemmas 13 and 14, we have the following theorem.Proof.Let A, F 1 and F 2 be defined in Lemma 2. By the Lemma 2, Suppose, on the contrary, that B n − F 1 − F 2 has at least one isolated vertex w.Since F 1 is a 2-good neighbor faulty set, there are two vertices u, v ∈ F 2 \ F 1 such that u and v are adjacent to w.Since the vertex set pair (F 1 , F 2 ) is not satisfied with any one condition in Lemma 10, this is a contradiction.Therefore, BS n − F 1 − F 2 has no isolated vertex.The proof of Claim 1 is complete. Let Since the vertex set pair (F 1 , F 2 ) is not satisfied with any one condition in Theorem 10, by the condition (1) of Lemma 10, for any pair of adjacent vertices u, w ∈ Since both F 1 and F 2 are 2-good-neighbor faulty sets, and there is no edge between Combining Lemmas 15 and 16, we have the following theorem.We point out that B 4 is the least bubble-sort graph satisfying the three sufficient conditions in Lemma 10.Because B 3 is a cycle with six vertices which is isomorphic to the 3-dimensional star graph, by [21]  Proof.By the definition of 3-good-neighbor diagnosability, it is sufficient to show that B n is 3-good-neighbor (8n − 25)-diagnosable.By Lemma 10, suppose, on the contrary, that there are two distinct 3-good-neighbor faulty subsets F 1 and F 2 of B n with |F 1 | ≤ 8n − 25 and |F 2 | ≤ 8n − 25, but the vertex set pair (F 1 , F 2 ) is not satisfied with any one condition in Lemma 10.Without loss of generality, assume that F 2 \ F 1 = ∅.Similarly to the discussion on V(B n ) = F 1 ∪ F 2 in Lemma 9, we have Claim 1. B n − F 1 − F 2 has no isolated vertex.Suppose, on the contrary, that B n − F 1 − F 2 has at least one isolated vertex w.Since F 1 is a 3-good neighbor faulty set, there are three vertices u, v ∈ F 2 \ F 1 such that u, v and x are adjacent to w.Since the vertex set pair (F 1 , F 2 ) is not satisfied with any one condition in Lemma 10, this is a contradiction.Therefore, BS n − F 1 − F 2 has no isolated vertex.The proof of Claim 1 is complete.
Let u ∈ V(B n ) \ (F 1 ∪ F 2 ).By Claim 1, u has at least one neighbor in B n − F 1 − F 2 .Since the vertex set pair (F 1 , F 2 ) is not satisfied with any one condition in Theorem 10, by the condition (1) of Lemma 10, for any pair of adjacent vertices u, w ∈ V(B n ) \ (F 1 ∪ F 2 ), there is no vertex v ∈ F 1 F 2 such that uw ∈ E(B n ) and vw ∈ E(B n ).It follows that u has no neighbor in F 1 F 2 .By the arbitrariness of u, there is no edge between V(B n ) \ (F 1 ∪ F 2 ) and F 1 F 2 .Since F 2 \ F 1 = ∅ and F 1 is a 3-good-neighbor faulty set, δ B n ([F 2 \ F 1 ]) ≥ 3.By Lemma 8, |F 2 \ F 1 | ≥ 8. Since both F 1 and F 2 are 3-good-neighbor faulty sets, and there is no edge between V(B n ) \ (F 1 ∪ F 2 ) and F 1 F 2 , F 1 ∩ F 2 is a 3-good-neighbor cut of B n .By Theorem 5, we have

Conclusions
In this paper, we investigate the problem of g-good-neighbor diagnosability of the n-dimensional bubble-sort graph B n under the PMC model and MM * model and show g-good-neighbor diagnosability of B n is 2 g (n − g) − 1 under the PMC model for g = 0, 1, 2, 3 and the MM * model for g = 0, 1, 2, 3, respectively.The work will help engineers to develop more different networks.
Author Contributions: S.W. and Z.W. conceived and designed the study and wrote the manuscript.S.W. revised the manuscript.All authors read and approved the final manuscript.

Figure 2 .Theorem 6 .
Figure 2. Illustration of a distinguishable pair (F 1 , F 2 ) under Preparata, Metze, and Chien's (PMC) model.Theorem 6.The diagnosability of the bubble-sort graph B n is n − 1 under the PMC model when n ≥ 4.

Lemma 4 .Lemma 5 .
Let H be a subgraph of B n such that δ(H) = 2. Then |V(H)| ≥ 4. By the definition of B n , we have Lemma 4. Let n ≥ 4. Then the 2-good-neighbor diagnosability t 2 (B n ) ≥ 4n − 9 under the PMC model.

Theorem 12 .
Let n ≥ 4. Then the 2-good-neighbor diagnosability of the bubble-sort star graph B n under the MM * model is 4n − 9.
2, • • • , n}, and let S n be the symmetric group on [n] containing all permutations p For any integer n ≥ 2, B n is bipartite.For any integer n ≥ 3, the girth of B n is 4.
Yuan et al. [26]presented a sufficient and necessary condition for a 94 system to be g-good-neighbor t-diagnosable under the PMC model.95Lemma 1. ([26]) A system G = (V, E) is g-good-neighbor t-diagnosable under the PMC model if and only if 96there 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 97 subsets F 1 and F 2 of V with |F 1 | ≤ t and |F 2 | ≤ t (See Fig.2).The g-good-neighbor diagnosability t g (G) of G 98 is the maximum value of t such that G is g-good-neighbor t-diagnosable under the PMC model.99Figure1.The bubble-sort graphs B 2 , B 3 and B 4 .Note that B n is a subclass of Cayley graphs.B n has the following useful properties.Proposition 1.For any integer n ≥ 2, B n is (n − 1)-regular and vertex transitive.Proposition 2.
The diagnosability of the bubble-sort graph B n is n − 1 under the PMC model when n ≥ 4.
101Proof.Let A and F 1 F 2 .By Lemma 1, we show that B n is not n-diagnosable under the PMC 104 model.Hence, by the definition of the diagnosability, we have that the diagnosability of B n is less than 105 n-diagnosable, i.e., t(B n ) = t 0 (B n ) ≤ n − 1.
(34)}, we have that B n [A] is a 4-cycle.By Propositions 3 and 4, |N B n (A)| = 4n − 12. Thus from calculating, we have |F 1 (12 F 1 is a 1-good-neighbor cut of B n .Since {(1),(12)} = F 1 F 2 337 and F 1 ⊂ F 2 , there is no edge of B n between V(B n )\(F 1 ∪ F 2 ) and F 1 F 2 .By Lemma 10, we show that 338 B n is not 1-good-neighbor (2n − 2)-diagnosable under the MM * model.Hence, by the definition of the 339 1-good-neighbor diagnosability, we have that t 1 (B n ) ≤ 2n − 3. Let n ≥ 5. Then the 1-good-neighbor diagnosability of the bubble-sort graph B n under the MM * 341 model is more than or equal to 2n − 3, i.e., t 1 (B n ) ≥ 2n − 3.By the definition of 1-good-neighbor diagnosability, it is sufficient to show that B n is Similarly to the discussion on V(B n ) = F 1 ∪ F 2 in Theorem 3, we 347 Figure Illustration of a distinguishable pair (F 1 , F ) under Maeng and Malek's (MM)* model.
and δ(B n − F 2 ) ≥ 2. So both F 1 and F 2 are 2-good-neighbor faulty sets.By the definitions of F 1 and F both F 1 and F 2 are not satisfied with any one condition in Lemma 10, and B n is not 2-good-neighbor (4n − 8)-diagnosable.Hence, t 2 (B n ) ≤ 4n − 9.The proof is complete.Let n ≥ 4. Then the diagnosability t 2 (B n ) ≥ 4n − 9 under the MM * model.Proof.By the definition of 2-good-neighbor diagnosability, it is sufficient to show that B n is 2-good-neighbor (4n − 9)-diagnosable.By Lemma 10, suppose, on the contrary, that there are two distinct 2-good-neighbor faulty subsets F 1 and F 2 of B n with |F 1 | ≤ 4n − 9 and |F 2 | ≤ 4n − 9, but the vertex set pair (F 1 , F 2 ) is not satisfied with any one condition in Lemma 10.Without loss of generality, assume that F 2