Open Access
This article is

- freely available
- re-usable

*Information*
**2019**,
*10*(1),
21;
https://doi.org/10.3390/info10010021

Article

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

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China

^{*}

Author to whom correspondence should be addressed.

Received: 2 December 2018 / Accepted: 4 January 2019 / Published: 10 January 2019

## Abstract

**:**

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 ${B}_{n}$ has many good properties. In this paper, we prove that (1) the 1-good-neighbor diagnosability of ${B}_{n}$ is $2n-3$ under Preparata, Metze, and Chien’s (PMC) model for $n\ge 4$ and Maeng and Malek’s (MM)${}^{\ast}$ model for $n\ge 5$; (2) the 2-good-neighbor diagnosability of ${B}_{n}$ is $4n-9$ under the PMC model and the MM${}^{\ast}$ model for $n\ge 4$; (3) the 3-good-neighbor diagnosability of ${B}_{n}$ is $8n-25$ under the PMC model and the MM${}^{\ast}$ model for $n\ge 7$.

Keywords:

interconnection network; graph; diagnosability; PMC model; MM* model; bubble-sort graph## 1. 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 $t\left(G\right)$ 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 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].

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 $n\ge 4$; (2) the 1-good-neighbor diagnosability of ${B}_{n}$ is $2n-3$ under the PMC model for $n\ge 4$ and the MM${}^{\ast}$ model for $n\ge 5$; (3) the 2-good-neighbor diagnosability of ${B}_{n}$ is $4n-9$ under the PMC model and the $M{M}^{\ast}$ model for $n\ge 4$; (4) the 3-good-neighbor diagnosability of ${B}_{n}$ is $8n-25$ under the PMC model and the MM${}^{\ast}$ model for $n\ge 7$.

## 2. Preliminaries

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

#### 2.1. 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 ${V}^{\prime}$ of V, the induced subgraph by ${V}^{\prime}$ in G, denoted by $G\left[{V}^{\prime}\right]$, is a graph, whose vertex set is ${V}^{\prime}$ and the edge set is the set of all the edges of G with both endpoints in ${V}^{\prime}$. The degree ${d}_{G}\left(v\right)$ of a vertex v is the number of edges incident with v. We denote by $\delta \left(G\right)$ the minimum degrees of vertices of G. For any vertex v, we define the neighborhood ${N}_{G}\left(v\right)$ 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 $u\in {N}_{G}\left(v\right)$. Let $S\subseteq V$. We use ${N}_{G}\left(S\right)$ to denote the set ${\cup}_{v\in S}{N}_{G}\left(v\right)\backslash S$. 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, ${d}_{G}\left(v\right)=k$. 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 $\left|X\right|=n$ and $\left|Y\right|=m$. Let $G=(V,E)$ be a connected graph. The connectivity $\kappa \left(G\right)$ 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 $F\subseteq V$ is called a g-good-neighbor faulty set if $\left|N\right(v)\cap (V\backslash F\left)\right|\ge g$ for every vertex v in $V\backslash F$. 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 ${\kappa}^{\left(g\right)}\left(G\right)$. For graph-theoretical terminology and notation not defined here we follow [22].

#### 2.2. 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.

In the permutation $\left(\genfrac{}{}{0pt}{}{1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n}{\phantom{\rule{4pt}{0ex}}{p}_{1}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{p}_{2}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{p}_{n}}\right)$, $i\u27f6{p}_{i}$. For the convenience, we denote the permutation $\left(\genfrac{}{}{0pt}{}{1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n}{\phantom{\rule{4pt}{0ex}}{p}_{1}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{p}_{2}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{p}_{n}}\right)$ by ${p}_{1}\phantom{\rule{4pt}{0ex}}{p}_{2}\phantom{\rule{4pt}{0ex}}\dots \phantom{\rule{4pt}{0ex}}{p}_{n}$. Every permutation can be denoted by a product of cycles [23]. For example, $\left(\genfrac{}{}{0pt}{}{1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}3}{3\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}2}\right)=\left(132\right)$. Specially, $\left(\genfrac{}{}{0pt}{}{1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n}{1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dots \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n}\right)=\left(1\right)$. The product $\sigma \tau $ of two permutations is the composition function $\tau $ followed by $\sigma $, for example, $\left(12\right)\left(13\right)=\left(132\right)$. For terminology and notation not defined here we follow [23].

Let $\left[n\right]=\{1,2,\cdots ,n\}$, and let ${S}_{n}$ be the symmetric group on $\left[n\right]$ containing all permutations $p={p}_{1}{p}_{2}\cdots {p}_{n}$ of $\left[n\right]$. It is well known that $\left\{\right(i,i+1):i=1,2,\dots ,n-1\}$ is a generating set for ${S}_{n}$. The n-dimensional bubble-sort graph ${B}_{n}$ [24] is the graph with vertex set $V\left({B}_{n}\right)$ = ${S}_{n}$ in which two vertices u, v are adjacent if and only if $u=v(i,i+1)$, $1\le i\le n-1$. It is easy to see from the definition that ${B}_{n}$ is a $(n-1)$-regular graph on $n!$ vertices. The graphs ${B}_{2}$, ${B}_{3}$ and ${B}_{4}$ are depicted in Figure 1.

Note that ${B}_{n}$ is a subclass of Cayley graphs. ${B}_{n}$ has the following useful properties.

**Proposition**

**1.**

For any integer $n\ge 2$, ${B}_{n}$ is $(n-1)$-regular and vertex transitive.

**Proposition**

**2.**

For any integer $n\ge 2$, ${B}_{n}$ is bipartite.

**Proposition**

**3.**

For any integer $n\ge 3$, the girth of ${B}_{n}$ is 4.

**Theorem**

**1**

([23])

**.**Every nonidentity permutation in the symmetric group is uniquely (up to the order of the factors) a product of disjoint cycles, each of which has length of at least 2.**Proposition**

**4**

([12])

**.**Let ${B}_{n}$ be a bubble-sort graph. If two vertices $u,v$ are adjacent, there is no common neighbor vertex of these two vertices, i.e., $\left|N\right(u)\cap N(v\left)\right|=0$. If two vertices $u,v$ are not adjacent, there is at most two common neighbor vertices of these two vertices, i.e., $\left|N\right(u)\cap N(v\left)\right|\le 2$.**Theorem**

**2**

**Theorem**

**5**

([27])

**.**${\kappa}^{\left(3\right)}\left({B}_{n}\right)=12$ for $n=5$ and ${\kappa}^{\left(3\right)}\left({B}_{n}\right)=8n-32$ for $n\ge 6$.## 3. 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 ${F}_{1}\Delta {F}_{2}=({F}_{1}\backslash {F}_{2})\cup ({F}_{2}\backslash {F}_{1})$. Yuan et al. [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 $uv\in E$ with $u\in V\backslash ({F}_{1}\cup {F}_{2})$ and $v\in {F}_{1}\Delta {F}_{2}$ for each distinct pair of g-good-neighbor faulty subsets ${F}_{1}$ and ${F}_{2}$ of V with $|{F}_{1}|\le t$ and $|{F}_{2}|\le t$ (See Figure 2). The g-good-neighbor diagnosability ${t}_{g}\left(G\right)$ of G is the maximum value of t such that G is g-good-neighbor t-diagnosable under the PMC model.**Theorem**

**6.**

The diagnosability of the bubble-sort graph ${B}_{n}$ is $n-1$ under the PMC model when $n\ge 4$.

**Proof.**

Let $A=\left\{\right(1\left)\right\}$. Then $\left|N\right(A\left)\right|=n-1$. Let ${F}_{1}=N\left(A\right)$ and ${F}_{2}=A\cup N\left(A\right)$. Then $|{F}_{1}|=n-1$ and $|{F}_{2}|=n$. Since $\left(1\right)={F}_{1}\Delta {F}_{2}$ and ${N}_{{B}_{n}}\left(\left(1\right)\right)={F}_{1}\subset {F}_{2}$, there is no edge of ${B}_{n}$ between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. 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 n-diagnosable, i.e., $t\left({B}_{n}\right)={t}_{0}\left({B}_{n}\right)\le n-1$.

By the definition of the diagnosability, it is sufficient to show that ${B}_{n}$ is $(n-1)$-diagnosable under the PMC model. By Lemma 1, to prove that ${B}_{n}$ is $(n-1)$-diagnosable, it is equivalent to prove that there is an edge $uv\in E\left({B}_{n}\right)$ with $u\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and $v\in {F}_{1}\Delta {F}_{2}$ for each distinct pair of faulty subsets ${F}_{1}$ and ${F}_{2}$ of $V\left({B}_{n}\right)$ with $|{F}_{1}|\le n-1$ and $|{F}_{2}|\le n-1$. We prove this statement by contradiction. Suppose that there are two distinct faulty subsets ${F}_{1}$ and ${F}_{2}$ of $V\left({B}_{n}\right)$ with $|{F}_{1}|\le n-1$ and $|{F}_{2}|\le n-1$, 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 $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. Without loss of generality, assume that ${F}_{2}\backslash {F}_{1}\ne \varnothing $. Suppose $V\left({B}_{n}\right)={F}_{1}\cup {F}_{2}$. By the definition of ${B}_{n}$, $|{F}_{1}\cup {F}_{2}|=|{S}_{n}|=n!$. It is obvious that $n!>2n-2$ for $n\ge 4$. Since $n\ge 4$, we have that $n!=|V\left({B}_{n}\right)|=|{F}_{1}\cup {F}_{2}|=|{F}_{1}|+|{F}_{2}|-|{F}_{1}\cap {F}_{2}|\le |{F}_{1}|+|{F}_{2}|<2n-2$, a contradiction. Therefore, $V\left({B}_{n}\right)\ne {F}_{1}\cup {F}_{2}$. Since there are no edges between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$, and $|V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})|\ne 0$ and $|{F}_{1}\Delta {F}_{2}|\ne 0$, we have that ${F}_{1}\cap {F}_{2}$ is a cut set. By Theorem 2, $|{F}_{1}\cap {F}_{2}|\ge n-1$. Therefore, $|{F}_{2}|=|{F}_{2}\backslash {F}_{1}|+|{F}_{1}\cap {F}_{2}|\ge 1+n-1=n$, which contradicts with that $|{F}_{2}|\le n-1$. So ${B}_{n}$ is $(n-1)$-diagnosable. By the definition of $t\left({B}_{n}\right)$, the diagnosability $t\left({B}_{n}\right)\ge n-1$. □

**Theorem**

**7.**

The 1-good-neighbor diagnosability of ${B}_{n}$ is $2n-3$ under the PMC model when $n\ge 4$.

**Proof.**

Let $A=\left\{\right(1),(12\left)\right\}$. By Proposition 2, $\left|N\right(A\left)\right|=2n-4$. Let ${F}_{1}=N\left(A\right)$ and ${F}_{2}=A\cup N\left(A\right)$. Then $|{F}_{1}|=2n-4$ and $|{F}_{2}|=2n-2$. Let $v\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$. By Proposition 4, $\left|N\right(v)\cap N(\left(1\right)\left)\right|\le 2$ and $\left|N\right(v)\cap N(\left(12\right)\left)\right|\le 2$. By Proposition 2, $N\left(v\right)\cap N\left(\right(1\left)\right)\ne \varnothing $ and $N\left(v\right)\cap N\left(\right(12\left)\right)=\varnothing $ or $N\left(v\right)\cap N\left(\right(12)\ne \varnothing $ and $N\left(v\right)\cap N\left(\right(1\left)\right)=\varnothing $. Therefore, $d\left(v\right)\ge n-1-2\ge 1$ ($n\ge 4$) in ${B}_{n}-({F}_{1}\cup {F}_{2})$ and ${F}_{1}$ is a 1-good-neighbor cut of ${B}_{n}$. Since $\{\left(1\right),\left(12\right)\}={F}_{1}\Delta {F}_{2}$ and ${F}_{1}\subset {F}_{2}$, there is no edge of ${B}_{n}$ between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. By Lemma 1, we show that ${B}_{n}$ is not 1-good-neighbor $(2n-2)$-diagnosable under the PMC model. Hence, by the definition of the 1-good-neighbor diagnosability, we have that ${t}_{1}\left({B}_{n}\right)\le 2n-3$.

By the definition of the 1-good-neighbor diagnosability, it is sufficient to show that ${B}_{n}$ is 1-good-neighbor $(2n-3)$-diagnosable. By Lemma 1, to prove that ${B}_{n}$ is 1-good-neighbor $(2n-3)$-diagnosable, it is equivalent to prove that there is an edge $uv\in E\left({B}_{n}\right)$ with $u\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and $v\in {F}_{1}\Delta {F}_{2}$ for each distinct pair of 1-good-neighbor faulty subsets ${F}_{1}$ and ${F}_{2}$ of $V\left({B}_{n}\right)$ with $|{F}_{1}|\le 2n-3$ and $|{F}_{2}|\le 2n-3$.

We prove this statement by contradiction. Suppose that there are two distinct 1-good-neighbor faulty subsets ${F}_{1}$ and ${F}_{2}$ of $V\left({B}_{n}\right)$ with $|{F}_{1}|\le 2n-3$ and $|{F}_{2}|\le 2n-3$, 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 $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. Without loss of generality, assume that ${F}_{2}\backslash {F}_{1}\ne \varnothing $. Suppose $V\left({B}_{n}\right)={F}_{1}\cup {F}_{2}$. Since $n\ge 4$, we have that $n!=|V\left({B}_{n}\right)|=|{F}_{1}\cup {F}_{2}|=|{F}_{1}|+|{F}_{2}|-|{F}_{1}\cap {F}_{2}|\le |{F}_{1}|+|{F}_{2}|\le 2(2n-3)=4n-6$, a contradiction. Therefore, $V\left({B}_{n}\right)\ne {F}_{1}\cup {F}_{2}$.

Since there are no edges between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$, and ${F}_{1}$ is a 1-good-neighbor faulty set, ${B}_{n}-{F}_{1}$ has two parts ${B}_{n}-{F}_{1}-{F}_{2}$ and ${B}_{n}[{F}_{2}\backslash {F}_{1}]$ (for convenience). Thus, $\delta ({B}_{n}-{F}_{1}-{F}_{2})\ge 1$ and $\delta \left({B}_{n}[{F}_{2}\backslash {F}_{1}]\right)\ge 1$. Similarly, $\delta \left({B}_{n}[{F}_{1}\backslash {F}_{2}]\right)\ge 1$ when ${F}_{1}\backslash {F}_{2}\ne \varnothing $. Therefore, ${F}_{1}\cap {F}_{2}$ is also a 1-good-neighbor faulty set. When ${F}_{1}\backslash {F}_{2}=\varnothing $, ${F}_{1}\cap {F}_{2}={F}_{1}$ is also a 1-good-neighbor faulty set. Since there are no edges between $V({B}_{n}-{F}_{1}-{F}_{2})$ and ${F}_{1}\Delta {F}_{2}$, ${F}_{1}\cap {F}_{2}$ is a 1-good-neighbor cut. By Theorem 3, $|{F}_{1}\cap {F}_{2}|\ge 2n-4$. Note that $|{F}_{2}\backslash {F}_{1}|\ge 2$. Therefore, $|{F}_{2}|=|{F}_{2}\backslash {F}_{1}|+|{F}_{1}\cap {F}_{2}|\ge 2+2n-4=2n-2$, which contradicts with that $|{F}_{2}|\le 2n-3$. So ${B}_{n}$ is 1-good-neighbor $(2n-3)$-diagnosable. By the definition of ${t}_{1}\left({B}_{n}\right)$, ${t}_{1}\left({B}_{n}\right)\ge 2n-3$. □

**Lemma**

**2.**

Let $A=\left\{\right(1),(12),(34),(12\left)\right(34\left)\right\}$. If $n\ge 4$, ${F}_{1}={N}_{{B}_{n}}\left(A\right)$, ${F}_{2}=A\cup {N}_{{B}_{n}}\left(A\right)$, then $|{F}_{1}|=4n-12$, $|{F}_{2}|=4n-8$, $\delta ({B}_{n}-{F}_{1})\ge 2$, and $\delta ({B}_{n}-{F}_{2})\ge 2$.

**Proof.**

By $A=\left\{\right(1),(12),(34),(12\left)\right(34\left)\right\}$, we have that ${B}_{n}\left[A\right]$ is a 4-cycle. By Propositions 3 and 4, $|{N}_{{B}_{n}}\left(A\right)|=4n-12$. Thus from calculating, we have $|{F}_{1}|=4n-12$, $|{F}_{2}|=|A|+|{F}_{1}|=4n-8$.

Let $v\in V\left({B}_{n}\right)\backslash {F}_{2}$ and $\left|N\right(v)\cap N(A\left)\right|\ne 0$ and $w\in N\left(v\right)\cap N\left(A\right)$. Let $u\in A$ and $uw\in E\left({B}_{n}\right)$. By Proposition 1, let $u=\left(1\right)$. Then $w=\left(ab\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(12),(34\left)\right\}$. By Proposition 2, there is no ${u}^{\prime}\in \{\left(12\right),\left(34\right)\}$ such that $|N\left({u}^{\prime}\right)\cap N\left(v\right)|\ge 1$. Therefore, we consider only ${u}^{\prime}\in \{\left(1\right),\left(12\right)\left(34\right)\}$. We discuss the following cases.

Case 1. $v=\left(ab\right)\left(cd\right)$ and $\{a,b\}\cap \{c,d\}=\varnothing $, $\left(cd\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(ab\left)\right\}$.

If $\left(cd\right)\in \left\{\right(12),(34\left)\right\}$, then a contradiction to $v\in V\left({B}_{n}\right)\backslash {F}_{2}$. Therefore, $\left(cd\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(ab),(12),(34\left)\right\}$. In this case, $\left|N\right(v)\cap N(u\left)\right|=2$. Consider $\left(12\right)\left(34\right)\left(xy\right)$ and $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(xy\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(12),(34\left)\right\}$. Suppose $\{x,y\}\cap \{1,2,3,4\}=\varnothing $. Since $\left(ab\right),\left(cd\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(12),(34\left)\right\}$, $\left(12\right)\left(34\right)\left(xy\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(xy\right)=\left(23\right)$, then $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)$. If $\left(uv\right)\ne \left(12\right)$, then, $1\to 1$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)=\left(12\right)$, then, $2\to 1$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(xy\right)=\left(45\right)$, then $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)$. If $\left(uv\right)\ne \left(12\right)$, then, $1\to 1$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)=\left(12\right)$, then, $3\to 3$ or $3\to 2$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. Therefore, $\left|N\right(v)\cap N(A\left)\right|\le 2$.

Case 2. $v=\left(ab\right)\left(cd\right)$ and $\{a,b\}\cap \{c,d\}\ne \varnothing $, $\left(cd\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(ab\left)\right\}$.

Without loss of generality, let $v=\left(ab\right)\left(bd\right)=\left(abd\right)$. Let ${w}^{\prime}\in N\left(v\right)\backslash \left\{w\right\}$. Then ${w}^{\prime}=\left(ab\right)\left(bd\right)\left(uv\right)$, $\left(uv\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(cd\left)\right\}$. If $\left(uv\right)=\left(ab\right)$, then ${w}^{\prime}=\left(ab\right)\left(bd\right)\left(uv\right)=\left(ad\right)$. Note $\left(ad\right)\notin \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}$. Then $\left|N\right(v)\cap N(u\left)\right|=1$. Suppose $\left(uv\right)\ne \left(ab\right)$. Consider $\left(12\right)\left(34\right)\left(xy\right)$ and $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(xy\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(12),(34\left)\right\}$. If $\{x,y\}\cap \{1,2,3,4\}=\varnothing $, then, by Theorem 1, $\left(12\right)\left(34\right)\left(xy\right)\ne {w}^{\prime}=\left(ab\right)\left(bd\right)\left(uv\right)$. If $\left(xy\right)=\left(23\right)$, then $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)$. If $\left(uv\right)=\left(12\right)$, then, $2\to 1$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)\ne \left(12\right)$, then, $1\to 1$ or $1\to 3$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(xy\right)=\left(45\right)$, then $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)$. If $\left(uv\right)\ne \left(12\right)$, then, $1\to 1$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)=\left(12\right)$, then, $3\to 3$ or $3\to 2$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. Therefore, $\left|N\right(v)\cap N(A\left)\right|\le 2$.

By Cases 1 and 2, $d\left(v\right)\ge n-1-2\ge 2$ ($n\ge 5$) in ${B}_{n}-({F}_{1}\cup {F}_{2})$ 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 $n\ge 4$. Then the 2-good-neighbor diagnosability ${t}_{2}\left({B}_{n}\right)\le 4n-9$ under the PMC model.

**Proof.**

Let A be defined in Lemma 2, and let ${F}_{1}={N}_{{B}_{n}}\left(A\right)$, ${F}_{2}=A\cup {N}_{{B}_{n}}\left(A\right)$. By Lemma 2, $|{F}_{1}|=4n-12$, $|{F}_{2}|=4n-8$, $\delta ({B}_{n}-{F}_{1})\ge 2$ and $\delta ({B}_{n}-{F}_{2})\ge 2$. Therefore, ${F}_{1}$ and ${F}_{2}$ are both 2-good-neighbor faulty sets of ${B}_{n}$ with $|{F}_{1}|=4n-12$ and $|{F}_{2}|=4n-8$. Since $A={F}_{1}\Delta {F}_{2}$ and ${N}_{{B}_{n}}\left(A\right)={F}_{1}\subset {F}_{2}$, there is no edge of ${B}_{n}$ between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. By Lemma 1, we show that ${B}_{n}$ is not 2-good-neighbor $(4n-8)$-diagnosable under the PMC model. Hence, by the definition of 2-good-neighbor diagnosability, we conclude that the 2-good-neighbor diagnosability of ${B}_{n}$ is less than $4n-8$, i.e., ${t}_{2}\left({B}_{n}\right)\le 4n-9$. □

**Lemma**

**4.**

Let H be a subgraph of ${B}_{n}$ such that $\delta \left(H\right)=2$. Then $\left|V\right(H\left)\right|\ge 4$.

By the definition of ${B}_{n}$, we have Lemma 4.

**Lemma**

**5.**

Let $n\ge 4$. Then the 2-good-neighbor diagnosability ${t}_{2}\left({B}_{n}\right)\ge 4n-9$ under the PMC 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 Theorem 1, to prove ${B}_{n}$ is 2-good-neighbor $(4n-9)$-diagnosable, it is equivalent to prove that there is an edge $uv\in E\left({B}_{n}\right)$ with $u\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and $v\in {F}_{1}\Delta {F}_{2}$ for each distinct pair of 2-good-neighbor faulty subsets ${F}_{1}$ and ${F}_{2}$ of $V\left({B}_{n}\right)$ with $|{F}_{1}|\le 4n-9$ and $|{F}_{2}|\le 4n-9$.

We prove this statement by contradiction. Suppose that there are two distinct 2-good-neighbor faulty subsets ${F}_{1}$ and ${F}_{2}$ of $V\left({B}_{n}\right)$ with $|{F}_{1}|\le 4n-9$ and $|{F}_{2}|\le 4n-9$, 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 $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. Without loss of generality, assume that ${F}_{2}\backslash {F}_{1}\ne \varnothing $. Suppose $V\left({B}_{n}\right)={F}_{1}\cup {F}_{2}$. By the definition of ${B}_{n}$, $|{F}_{1}\cup {F}_{2}|=|{S}_{n}|=n!$. It is obvious that $n!>8n-18$ for $n\ge 4$. Since $n\ge 4$, we have that $n!=|V\left({B}_{n}\right)|=|{F}_{1}\cup {F}_{2}|=|{F}_{1}|+|{F}_{2}|-|{F}_{1}\cap {F}_{2}|\le |{F}_{1}|+|{F}_{2}|\le 2(4n-9)=8n-18$, a contradiction. Therefore, $V\left({B}_{n}\right)\ne {F}_{1}\cup {F}_{2}$.

Since there are no edges between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$, and ${F}_{1}$ is a 2-good-neighbor faulty set, ${B}_{n}-{F}_{1}$ has two parts ${B}_{n}-{F}_{1}-{F}_{2}$ and ${B}_{n}[{F}_{2}\backslash {F}_{1}]$. Thus, $\delta ({B}_{n}-{F}_{1}-{F}_{2})\ge 2$ and $\delta \left({B}_{n}[{F}_{2}\backslash {F}_{1}]\right)\ge 2$. Similarly, $\delta \left({B}_{n}[{F}_{1}\backslash {F}_{2}]\right)\ge 2$ when ${F}_{1}\backslash {F}_{2}\ne \varnothing $. Therefore, ${F}_{1}\cap {F}_{2}$ is also a 2-good-neighbor faulty set. When ${F}_{1}\backslash {F}_{2}=\varnothing $, ${F}_{1}\cap {F}_{2}={F}_{1}$ is also a 2-good-neighbor faulty set. Since there are no edges between $V({B}_{n}-{F}_{1}-{F}_{2})$ and ${F}_{1}\Delta {F}_{2}$, ${F}_{1}\cap {F}_{2}$ is a 2-good-neighbor cut. Since $n\ge 4$, by Theorem 4, $|{F}_{1}\cap {F}_{2}|\ge 4n-12$. By Lemma 4, $|{F}_{2}\backslash {F}_{1}|\ge 4$. Therefore, $|{F}_{2}|=|{F}_{2}\backslash {F}_{1}|+|{F}_{1}\cap {F}_{2}|\ge 4+8n-22=8n-18$, which contradicts with that $|{F}_{2}|\le 4n-9$. So ${B}_{n}$ is 2-good-neighbor $(8n-19)$-diagnosable. By the definition of ${t}_{2}\left({B}_{n}\right)$, ${t}_{2}\left({B}_{n}\right)\ge 4n-9$. □

Combining Lemmas 3 and 5, we have the following theorem.

**Theorem**

**8.**

Let $n\ge 4$. Then the 2-good-neighbor diagnosability of the bubble-sort graph ${B}_{n}$ under the PMC model is $4n-9$.

**Lemma**

**6.**

Let $A=\left\{\right(1),(12),(34),(56),(12\left)\right(34),(12\left)\right(56),(34\left)\right(56),(12\left)\right(34\left)\right(56\left)\right\}$. If $n\ge 7$, ${F}_{1}={N}_{{B}_{n}}\left(A\right)$, ${F}_{2}=A\cup {N}_{{B}_{n}}\left(A\right)$, then $|{F}_{1}|=8n-32$, $|{F}_{2}|=8n-24$, $\delta ({B}_{n}-{F}_{1})\ge 3$ and $\delta ({B}_{n}-{F}_{2})\ge 3$.

**Proof.**

By $A=\left\{\right(1),(12),(34),(56),(12\left)\right(34),(12\left)\right(56),(34\left)\right(56),(12\left)\right(34\left)\right(56\left)\right\}$, we have that ${B}_{n}\left[A\right]$ is 3-regular and $\left|A\right|=8$.

Claim 1. $\left(N\right(u)\cap N(v\left)\right)\backslash A=\varnothing $ for $u,v\in A$.

By Proposition 1, let $u=\left(1\right)$. By Proposition 2, we consider only $v\in \left\{\right(12\left)\right(34),(12\left)\right(56),(34\left)\right(56\left)\right\}$. Since $\left|N\right(u)\cap N(v\left)\right|=2$, by Proposition 4, we have $\left(N\right(u)\cap N(v\left)\right)\backslash A=\varnothing $. The proof of Claim 1 is complete.

By Claim 1, $|{N}_{{B}_{n}}\left(A\right)|=8n-32$. Thus from calculating, we have $|{F}_{1}|=8n-32$, $|{F}_{2}|=|A|+|{F}_{1}|=8n-24$. Let $v\in V\left({B}_{n}\right)\backslash {F}_{2}$ and $\left|N\right(v)\cap N(A\left)\right|\ne 0$ and $w\in N\left(v\right)\cap N\left(A\right)$. Let $u\in A$ and $uw\in E\left({B}_{n}\right)$. By Proposition 1, let $u=\left(1\right)$. Then $w=\left(ab\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(12),(34),(56\left)\right\}$. By Proposition 2, there is no ${u}^{\prime}\in \{\left(12\right),\left(34\right),\left(56\right),\left(12\right)\left(34\right)\left(56\right)\}$ such that $|N\left({u}^{\prime}\right)\cap N\left(v\right)|\ge 1$. Therefore, we consider only ${u}^{\prime}\in \{\left(1\right),\left(12\right)\left(34\right),\left(12\right)\left(56\right),\left(34\right)\left(56\right)\}$.

Claim 2. $\left|N\right(A)\cap N(v\left)\right|\le 2$.

Let $v\in V\left({B}_{n}\right)\backslash {F}_{2}$. We discuss the following cases.

Case 1.$v=\left(ab\right)\left(cd\right)$ and $\{a,b\}\cap \{c,d\}=\varnothing $, $\left(cd\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(ab\left)\right\}$.

If $\left(cd\right)\in \left\{\right(12),(34),(56\left)\right\}$, then a contradiction to $v\in V\left({B}_{n}\right)\backslash {F}_{2}$. Therefore, $\left(cd\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(ab),(12),(34),(56\left)\right\}$. Consider $\left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)=\left(ab\right)$, then $\left|N\right(v)\cap N(u\left)\right|=2$. Let $\left(uv\right)\ne \left(ab\right)$.

Consider $\left(12\right)\left(34\right)\left(xy\right)$ and $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(xy\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(12),(34),(56\left)\right\}$. Suppose $\{x,y\}\cap \{1,2,3,4\}=\varnothing $. Since $\left(ab\right),\left(cd\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(12),(34),(56\left)\right\}$, $\left(12\right)\left(34\right)\left(xy\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(xy\right)=\left(23\right)$, then $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)$. If $\left(uv\right)\ne \left(12\right)$, then, $1\to 1$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)=\left(12\right)$, then, $2\to 1$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(xy\right)=\left(45\right)$, then $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)$. If $\left(uv\right)\ne \left(12\right)$, then, $1\to 1$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)=\left(12\right)$, then, $3\to 3$ or $3\to 2$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$.

Consider $\left(34\right)\left(56\right)\left(xy\right)$ and $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(xy\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(12),(34),(56\left)\right\}$. Suppose $\{x,y\}\cap \{3,4,5,6\}=\varnothing $. Since $\left(ab\right),\left(cd\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(12),(34),(56\left)\right\}$, $\left(34\right)\left(56\right)\left(xy\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(xy\right)=\left(23\right)$, then $\left(34\right)\left(23\right)\left(56\right)=\left(243\right)\left(56\right)$. If $\left(uv\right)=\left(23\right)$, then $2\to 3$ or $2\to 2$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(34\right)\left(23\right)\left(56\right)=\left(243\right)\left(56\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)\ne \left(23\right)$, then $\left(uv\right)=\left(12\right)$ or $\left(34\right)$ or $\left(uv\right)$ ($u,v\ge 4$). When $\left(uv\right)=\left(12\right)$, $2\to 1$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(34\right)\left(23\right)\left(56\right)=\left(243\right)\left(56\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. When $\left(uv\right)=\left(34\right)$, $2\to 2$ or $2\to 3$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(34\right)\left(23\right)\left(56\right)=\left(243\right)\left(56\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. When $u,v\ge 4$, $2\to 2$ or $2\to 3$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(34\right)\left(23\right)\left(56\right)=\left(243\right)\left(56\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$.

Similarly, consider $\left(12\right)\left(56\right)\left(xy\right)$ and $\left(ab\right)\left(cd\right)\left(uv\right)$. We have $\left(12\right)\left(56\right)\left(xy\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. Therefore, $\left|N\right(v)\cap N(A\left)\right|\le 2$.

Case 2.$v=\left(ab\right)\left(cd\right)$ and $\{a,b\}\cap \{c,d\}\ne \varnothing $, $\left(cd\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(ab\left)\right\}$.

Without loss of generality, let $v=\left(ab\right)\left(bd\right)=\left(abd\right)$. Let ${w}^{\prime}\in N\left(v\right)\backslash \left\{w\right\}$. Then ${w}^{\prime}=\left(ab\right)\left(bd\right)\left(uv\right)$, $\left(uv\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(cd\left)\right\}$. If $\left(uv\right)=\left(ab\right)$, then ${w}^{\prime}=\left(ab\right)\left(bd\right)\left(uv\right)=\left(ad\right)$. Note $\left(ad\right)\notin \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}$. Then $\left|N\right(v)\cap N(u\left)\right|=1$. Suppose $\left(uv\right)\ne \left(ab\right)$. Consider $\left(12\right)\left(34\right)\left(xy\right)$ and $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(xy\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(12),(34),(56\left)\right\}$. If $\{x,y\}\cap \{1,2,3,4\}=\varnothing $, then, by Theorem 1, $\left(12\right)\left(34\right)\left(xy\right)\ne {w}^{\prime}=\left(ab\right)\left(bd\right)\left(uv\right)$. If $\left(xy\right)=\left(23\right)$, then $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)$. If $\left(uv\right)=\left(12\right)$, then, $2\to 1$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)\ne \left(12\right)$, then, $1\to 1$ or $1\to 3$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(23\right)=\left(1243\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(xy\right)=\left(45\right)$, then $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)$. If $\left(uv\right)\ne \left(12\right)$, then, $1\to 1$ or $1\to 3$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)=\left(12\right)$, then, $3\to 3$ or $3\to 2$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(12\right)\left(34\right)\left(45\right)=\left(12\right)\left(345\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$.

Consider $\left(34\right)\left(56\right)\left(xy\right)$ and $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(xy\right)\in \left\{\right(i,i+1):i=1,2,3,\dots ,n-1\}\backslash \left\{\right(12),(34),(56\left)\right\}$. If $\{x,y\}\cap \{3,4,5,6\}=\varnothing $, then, by Theorem 1, $\left(34\right)\left(56\right)\left(xy\right)\ne {w}^{\prime}=\left(ab\right)\left(bd\right)\left(uv\right)$. If $\left(xy\right)=\left(23\right)$, then $\left(34\right)\left(56\right)\left(xy\right)=\left(243\right)\left(56\right)$. If $\left(uv\right)=\left(12\right)$, then, $2\to 1$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(34\right)\left(56\right)\left(xy\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)\ne \left(12\right)$, then, $1\to 1$ or $1\to 3$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(34\right)\left(56\right)\left(xy\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(xy\right)=\left(45\right)$, then $\left(34\right)\left(56\right)\left(xy\right)=\left(3465\right)$. If $\left(uv\right)\ne \left(12\right)$, then, $1\to 1$ or $1\to 3$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(34\right)\left(56\right)\left(xy\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)=\left(12\right)$, then, $3\to 3$ or $3\to 2$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(34\right)\left(56\right)\left(xy\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(xy\right)=\left(67\right)$, then $\left(34\right)\left(56\right)\left(xy\right)=\left(34\right)\left(567\right)$. If $\left(uv\right)\ne \left(12\right)$, then, $1\to 1$ or $1\to 3$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(34\right)\left(56\right)\left(xy\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. If $\left(uv\right)=\left(12\right)$, then, $3\to 3$ or $3\to 2$ in $\left(ab\right)\left(cd\right)\left(uv\right)$, $\left(34\right)\left(56\right)\left(xy\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$.

Similarly, consider $\left(12\right)\left(56\right)\left(xy\right)$ and $\left(ab\right)\left(cd\right)\left(uv\right)$. We have $\left(12\right)\left(56\right)\left(xy\right)\ne \left(ab\right)\left(cd\right)\left(uv\right)$. Therefore, $\left|N\right(v)\cap N(A\left)\right|\le 2$. The proof of Claim 2 is complete.

By Claim 2, $d\left(v\right)\ge n-1-3\ge 3$ ($n\ge 7$) in ${B}_{n}-({F}_{1}\cup {F}_{2})$ and ${F}_{1}$ is a 3-good-neighbor cut of ${B}_{n}$. □

**Lemma**

**7.**

Let $n\ge 7$. Then the 3-good-neighbor diagnosability ${t}_{3}\left({B}_{n}\right)\le 8n-25$ under the PMC model.

**Proof.**

Let A be defined in Lemma 6, and let ${F}_{1}={N}_{{B}_{n}}\left(A\right)$, ${F}_{2}=A\cup {N}_{{B}_{n}}\left(A\right)$. By Lemma 6, $|{F}_{1}|=8n-32$, $|{F}_{2}|=8n-24$, $\delta ({B}_{n}-{F}_{1})\ge 3$ and $\delta ({B}_{n}-{F}_{2})\ge 3$. Therefore, ${F}_{1}$ and ${F}_{2}$ are both 3-good-neighbor faulty sets of ${B}_{n}$ with $|{F}_{1}|=8n-32$ and $|{F}_{2}|=8n-24$. Since $A={F}_{1}\Delta {F}_{2}$ and ${N}_{{B}_{n}}\left(A\right)={F}_{1}\subset {F}_{2}$, there is no edge of ${B}_{n}$ between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. 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., ${t}_{2}\left({B}_{n}\right)\le 8n-25$. □

**Lemma**

**8.**

Let H be a subgraph of ${B}_{n}$ such that $\delta \left(H\right)=3$. Then $\left|V\right(H\left)\right|\ge 8$.

**Proof.**

Note that there is no subgraph ${K}_{3,3}$ of ${B}_{n}$. Suppose, on the contrary, that there is a subgraph ${H}^{\prime}$ of ${B}_{n}$ such that $\delta \left({H}^{\prime}\right)\ge 3$ and $\left|V\right({H}^{\prime}\left)\right|=7$. Since ${B}_{n}$ is bipartite, let $V\left({H}^{\prime}\right)=(U,W)$ and $\left|U\right|=3$, $\left|W\right|=4$. By Proposition 1, let $W=\left\{\right(1),x,y,z\}$ and $U=\{a,b,c\}$. Since $\delta \left({H}^{\prime}\right)\ge 3$, $N\left(x\right)\cap N\left(y\right)=\{a,b,c\}$, a contradiction to Proposition 4. Therefore, $\left|V\right(H\left)\right|\ge 8$. □

**Lemma**

**9.**

Let $n\ge 7$. Then the 3-good-neighbor diagnosability ${t}_{3}\left({B}_{n}\right)\ge 8n-25$ under the PMC model.

**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 $uv\in E\left({B}_{n}\right)$ with $u\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and $v\in {F}_{1}\Delta {F}_{2}$ for each distinct pair of 3-good-neighbor faulty subsets ${F}_{1}$ and ${F}_{2}$ of $V\left({B}_{n}\right)$ with $|{F}_{1}|\le 8n-25$ and $|{F}_{2}|\le 8n-25$.

We prove this statement by contradiction. Suppose that there are two distinct 3-good-neighbor faulty subsets ${F}_{1}$ and ${F}_{2}$ of $V\left({B}_{n}\right)$ with $|{F}_{1}|\le 8n-25$ and $|{F}_{2}|\le 8n-25$, 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 $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. Without loss of generality, assume that ${F}_{2}\backslash {F}_{1}\ne \varnothing $. Suppose $V\left({B}_{n}\right)={F}_{1}\cup {F}_{2}$. By the definition of ${B}_{n}$, $|{F}_{1}\cup {F}_{2}|=|{S}_{n}|=n!$. It is obvious that $n!>16n-50$ for $n\ge 7$. Since $n\ge 7$, we have that $n!=|V\left({B}_{n}\right)|=|{F}_{1}\cup {F}_{2}|=|{F}_{1}|+|{F}_{2}|-|{F}_{1}\cap {F}_{2}|\le |{F}_{1}|+|{F}_{2}|\le 2(8n-25)=16n-50$, a contradiction. Therefore, $V\left({B}_{n}\right)\ne {F}_{1}\cup {F}_{2}$.

Since there are no edges between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$, and ${F}_{1}$ is a 3-good-neighbor faulty set, ${B}_{n}-{F}_{1}$ has two parts ${B}_{n}-{F}_{1}-{F}_{2}$ and ${B}_{n}[{F}_{2}\backslash {F}_{1}]$. Thus, $\delta ({B}_{n}-{F}_{1}-{F}_{2})\ge 3$ and $\delta \left({B}_{n}[{F}_{2}\backslash {F}_{1}]\right)\ge 3$. Similarly, $\delta \left({B}_{n}[{F}_{1}\backslash {F}_{2}]\right)\ge 3$ when ${F}_{1}\backslash {F}_{2}\ne \varnothing $. Therefore, ${F}_{1}\cap {F}_{2}$ is also a 3-good-neighbor faulty set. When ${F}_{1}\backslash {F}_{2}=\varnothing $, ${F}_{1}\cap {F}_{2}={F}_{1}$ is also a 3-good-neighbor faulty set. Since there are no edges between $V({B}_{n}-{F}_{1}-{F}_{2})$ and ${F}_{1}\Delta {F}_{2}$, ${F}_{1}\cap {F}_{2}$ is a 3-good-neighbor cut. Since $n\ge 7$, by Theorem 5, $|{F}_{1}\cap {F}_{2}|\ge 8n-32$. By Lemma 8, $|{F}_{2}\backslash {F}_{1}|\ge 8$. Therefore, $|{F}_{2}|=|{F}_{2}\backslash {F}_{1}|+|{F}_{1}\cap {F}_{2}|\ge 8+8n-32=8n-24$, which contradicts with that $|{F}_{2}|\le 8n-25$. So ${B}_{n}$ is 3-good-neighbor $(8n-25)$-diagnosable. By the definition of ${t}_{3}\left({B}_{n}\right)$, ${t}_{3}\left({B}_{n}\right)\ge 8n-25$. □

Combining Lemmas 7 and 9, we have the following theorem.

**Theorem**

**9.**

Let $n\ge 7$. Then the 3-good-neighbor diagnosability of the bubble-sort graph ${B}_{n}$ under the PMC model is $8n-25$.

## 4. The Diagnosability of the Bubble-Sort Graph ${B}_{n}$ under the MM${}^{\ast}$ Model

Before discussing the diagnosability of the bubble-sort graph ${B}_{n}$ under the MM${}^{\ast}$ model, we first give an existing result.

**Lemma**

**10**

([4,20])

**.**A system $G=(V,E)$ is g-good-neighbor t-diagnosable under the $M{M}^{\ast}$ model if and only if for each distinct pair of g-good-neighbor faulty subsets ${F}_{1}$ and ${F}_{2}$ of V with $|{F}_{1}|\le t$ and $|{F}_{2}|\le t$ satisfies one of the following conditions. (1) There are two vertices $u,w\in V\backslash ({F}_{1}\cup {F}_{2})$ and there is a vertex $v\in {F}_{1}\Delta {F}_{2}$ such that $uw\in E$ and $vw\in E$. (2) There are two vertices $u,v\in {F}_{1}\backslash {F}_{2}$ and there is a vertex $w\in V\backslash ({F}_{1}\cup {F}_{2})$ such that $uw\in E$ and $vw\in E$. (3) There are two vertices $u,v\in {F}_{2}\backslash {F}_{1}$ and there is a vertex $w\in V\backslash ({F}_{1}\cup {F}_{2})$ such that $uw\in E$ and $vw\in E$ (See Figure 3). The g-good-neighbor diagnosability ${t}_{g}\left(G\right)$ of G is the maximum value of t such that G is g-good-neighbor t-diagnosable under the MM${}^{\ast}$ model.**Theorem**

**10**

([12])

**.**The diagnosability $t\left(G\right)={t}_{0}\left(G\right)$ of ${B}_{n}$ is $n-1$ under the MM${}^{\ast}$ model when $n\ge 4$.A component of a graph G is odd according as it has an odd number of vertices. We denote by $o\left(G\right)$ the number of odd component of G.

**Lemma**

**11**

([22])

**.**A graph $G=(V,E)$ has a perfect matching if and only if $o(G-S)\le \left|S\right|$ for all $S\subseteq V$.**Lemma**

**12**

([22])

**.**Let $k\ge 0$ be an integer. Then every k-regular bipartite graph has k edge-disjoint perfect matchings.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 $n\ge 4$. Then the 1-good-neighbor diagnosability of the bubble-sort graph ${B}_{n}$ under the MM${}^{\ast}$ model is less than or equal to $2n-3$, i.e., ${t}_{1}\left({B}_{n}\right)\le 2n-3$.

**Proof.**

Let $u=\left(1\right)$ and $v=\left(12\right)$. Then u is adjacent to v. Let ${F}_{1}=N\left(\{u,v\}\right)$ and ${F}_{2}={F}_{1}\cup \{u,v\}$. By Proposition 2, $|{F}_{1}|=2n-4$, $|{F}_{2}|=2n-2$. Let $w\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$. By Proposition 4, $\left|N\right(w)\cap N(\left(1\right)\left)\right|\le 2$ and $\left|N\right(w)\cap N(\left(12\right)\left)\right|\le 2$. By Proposition 2, if $N\left(w\right)\cap N\left(\right(1\left)\right)\ne \varnothing $, then $N\left(w\right)\cap N\left(\right(12\left)\right)=\varnothing $ or if $N\left(w\right)\cap N\left(\right(12)\ne \varnothing $, then $N\left(w\right)\cap N\left(\right(1\left)\right)=\varnothing $. Therefore, $d\left(v\right)\ge n-1-2\ge 1$ ($n\ge 4$) in ${B}_{n}-({F}_{1}\cup {F}_{2})$ and ${F}_{1}$ is a 1-good-neighbor cut of ${B}_{n}$. Since $\{\left(1\right),\left(12\right)\}={F}_{1}\Delta {F}_{2}$ and ${F}_{1}\subset {F}_{2}$, there is no edge of ${B}_{n}$ between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. By Lemma 10, we show that ${B}_{n}$ is not 1-good-neighbor $(2n-2)$-diagnosable under the MM${}^{\ast}$ model. Hence, by the definition of the 1-good-neighbor diagnosability, we have that ${t}_{1}\left({B}_{n}\right)\le 2n-3$. □

**Lemma**

**14.**

Let $n\ge 5$. Then the 1-good-neighbor diagnosability of the bubble-sort graph ${B}_{n}$ under the MM${}^{\ast}$ model is more than or equal to $2n-3$, i.e., ${t}_{1}\left({B}_{n}\right)\ge 2n-3$.

**Proof.**

By the definition of 1-good-neighbor diagnosability, it is sufficient to show that ${B}_{n}$ is 1-good-neighbor $(2n-3)$-diagnosable. By Lemma 10, suppose, on the contrary, that there are two distinct 1-good-neighbor faulty subsets ${F}_{1}$ and ${F}_{2}$ of ${B}_{n}$ with $|{F}_{1}|\le 2n-3$ and $|{F}_{2}|\le 2n-3$, but the vertex set pair $({F}_{1},{F}_{2})$ is not satisfied with any one condition in Theorem 10. Without loss of generality, assume that ${F}_{2}\backslash {F}_{1}\ne \varnothing .$ Similarly to the discussion on $V\left({B}_{n}\right)\ne {F}_{1}\cup {F}_{2}$ in Theorem 3, we have $V\left({B}_{n}\right)\ne {F}_{1}\cup {F}_{2}$.

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 1-good-neighbor faulty set, there is a vertex $u\in {F}_{2}\backslash {F}_{1}$ such that u is adjacent to w. Since the vertex set pair $({F}_{1},{F}_{2})$ is not satisfied with any one condition in Lemma 10, there is at most one vertex $u\in {F}_{2}\backslash {F}_{1}$ such that u is adjacent to w. Thus, there is just a vertex $u\in {F}_{2}\backslash {F}_{1}$ such that u is adjacent to w. Assume ${F}_{1}\backslash {F}_{2}=\varnothing $. Then ${F}_{1}\subseteq {F}_{2}$. Since ${F}_{2}$ is a 1-good-neighbor faulty set, ${B}_{n}-{F}_{2}={B}_{n}-{F}_{1}-{F}_{2}$ has no isolated vertex, a contradiction. Therefore, let ${F}_{1}\backslash {F}_{2}\ne \varnothing $ as follows. Similarly, we can show that there is just a vertex $v\in {F}_{1}\backslash {F}_{2}$ such that v is adjacent to w. Let $W\subseteq V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ be the set of isolated vertices in ${B}_{n}[V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})]$, and let H be the subgraph induced by the vertex set $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2}\cup W)$. Then for any $w\in W$, there are $(n-3)$ neighbors in ${F}_{1}\cap {F}_{2}$. By Corollary 1, ${B}_{n}$ has a perfect matching. By Lemma 11, $\left|W\right|\le o(G-({F}_{1}\cup {F}_{2}))\le |{F}_{1}\cup {F}_{2}|\le |{F}_{1}|+|{F}_{2}|-|{F}_{1}\cap {F}_{2}|\le 2(2n-3)-(n-3)=3n-3$. Assume $V\left(H\right)=\varnothing $. Note that $n!=|V\left({B}_{n}\right)|=|{F}_{1}\cup {F}_{2}|+|W|\le 2(3n-3)=6n-6$. This is a contradiction to $n\ge 5$. So $V\left(H\right)\ne \varnothing $. Since the vertex set pair $({F}_{1},{F}_{2})$ is not satisfied with the condition (1) of Theorem 10, and any vertex of $V\left(H\right)$ is not isolated in H, we induce that there is no edge between $V\left(H\right)$ and ${F}_{1}\Delta {F}_{2}$. Thus, ${F}_{1}\cap {F}_{2}$ is a vertex cut of ${B}_{n}$ and $\delta ({B}_{n}-({F}_{1}\cap {F}_{2}))\ge 1$, i.e., ${F}_{1}\cap {F}_{2}$ is a 1-good-neighbor cut of ${B}_{n}$. By Theorem 3, $|{F}_{1}\cap {F}_{2}|\ge 2n-4$. Because $|{F}_{1}|\le 2n-3$ and $|{F}_{2}|\le 2n-3$, and neither ${F}_{1}\backslash {F}_{2}$ nor ${F}_{2}\backslash {F}_{1}$ is empty, we have $|{F}_{1}\backslash {F}_{2}|=|{F}_{2}\backslash {F}_{1}|=1$. Let ${F}_{1}\backslash {F}_{2}=\left\{{v}_{1}\right\}$ and ${F}_{2}\backslash {F}_{1}=\left\{{v}_{2}\right\}$. Then for any vertex $w\in W$, w is adjacent to ${v}_{1}$ and ${v}_{2}$. According to Proposition 4, there are at most three common neighbors for any pair of vertices in ${B}_{n}$, it follows that there are at most two isolated vertices in ${B}_{n}-{F}_{1}-{F}_{2}$, i.e., $\left|W\right|\le 2$.

Suppose that there is exactly one isolated vertex v in ${B}_{n}-{F}_{1}-{F}_{2}$. Let ${v}_{1}$ and ${v}_{2}$ be adjacent to v. Then ${N}_{{B}_{n}}\left(v\right)\backslash \{{v}_{1},{v}_{2}\}\subseteq {F}_{1}\cap {F}_{2}$. Note that ${B}_{n}$ has no 3-cycle. Thus, ${N}_{{B}_{n}}\left({v}_{1}\right)\backslash \left\{v\right\}\subseteq {F}_{1}\cap {F}_{2}$, ${N}_{{B}_{n}}\left({v}_{2}\right)\backslash \left\{v\right\}\subseteq {F}_{1}\cap {F}_{2}$, $|({N}_{{B}_{n}}\left(v\right)\backslash \{{v}_{1},{v}_{2}\})\cap ({N}_{{B}_{n}}\left({v}_{1}\right)\backslash \left\{v\right\})|=0$ and $|({N}_{{B}_{n}}\left(v\right)\backslash \{{v}_{1},{v}_{2}\})\cap ({N}_{{B}_{n}}\left({v}_{2}\right)\backslash \left\{v\right\})|=0$ and $|[{N}_{{B}_{n}}\left({v}_{1}\right)\backslash \left\{v\right\}]\cap [{N}_{{B}_{n}}\left({v}_{2}\right)\backslash \left\{v\right\}]|\le 1$. Thus, $|{F}_{1}\cap {F}_{2}|\ge |{N}_{{B}_{n}}\left(v\right)\backslash \{{v}_{1},{v}_{2}\}|+|{N}_{{B}_{n}}\left({v}_{1}\right)\backslash \left\{v\right\}|+|{N}_{{B}_{n}}\left({v}_{2}\right)\backslash \left\{v\right\}|\ge (n-1-2)+(n-1-1)+(n-1-1)-1=3n-8$. It follows that $|{F}_{2}|=|{F}_{2}\backslash {F}_{1}|+|{F}_{1}\cap {F}_{2}|\ge 1+3n-8=3n-7>2n-3\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}(n\ge 5)$, which contradicts $|{F}_{2}|\le 2n-3$.

Suppose that there are exactly two isolated vertices v and w in ${B}_{n}-{F}_{1}-{F}_{2}$. Let ${v}_{1}$ and ${v}_{2}$ be adjacent to v and w, respectively. Then ${N}_{{B}_{n}}\left(v\right)\backslash \{{v}_{1},{v}_{2}\}\subseteq {F}_{1}\cap {F}_{2}$, ${N}_{{B}_{n}}\left(w\right)\backslash \{{v}_{1},{v}_{2}\}\subseteq {F}_{1}\cap {F}_{2}$, ${N}_{{B}_{n}}\left({v}_{1}\right)\backslash \{v,w\}\subseteq {F}_{1}\cap {F}_{2}$, ${N}_{{B}_{n}}\left({v}_{2}\right)\backslash \{v,w\}\subseteq {F}_{1}\cap {F}_{2}$, $|({N}_{{B}_{n}}\left(v\right)\backslash \{{v}_{1},{v}_{2}\})\cap ({N}_{{B}_{n}}\left({v}_{1}\right)\backslash \{v,w\})|=0$ and $|({N}_{{B}_{n}}\left(v\right)\backslash \{{v}_{1},{v}_{2}\})\cap ({N}_{{B}_{n}}\left({v}_{2}\right)\backslash \{v,w\})|=0$. $|({N}_{{B}_{n}}\left(w\right)\backslash \{{v}_{1},{v}_{2}\})\cap ({N}_{{B}_{n}}\left({v}_{1}\right)\backslash \{v,w\})|=0$, $|({N}_{{B}_{n}}\left(w\right)\backslash \{{v}_{1},{v}_{2}\})\cap ({N}_{{B}_{n}}\left({v}_{2}\right)\backslash \{v,w\})|=0$ and $|[{N}_{{B}_{n}}\left({v}_{1}\right)\backslash \{v,w\}]\cap [{N}_{{B}_{n}}\left({v}_{2}\right)\backslash \{v,w\}]|=0$. By Proposition 4, there are at most two common neighbors for any pair of vertices in ${B}_{n}$. Thus, it follows that $|({N}_{{B}_{n}}\left(v\right)\backslash \{{v}_{1},{v}_{2}\})\cap ({N}_{{B}_{n}}\left(w\right)\backslash \{{v}_{1},{v}_{2}\})|=0$. Thus, $|{F}_{1}\cap {F}_{2}|\ge |{N}_{{B}_{n}}\left(v\right)\backslash \{{v}_{1},{v}_{2}\}|+|{N}_{{B}_{n}}\left(w\right)\backslash \{{v}_{1},{v}_{2}\}|+|{N}_{{B}_{n}}\left({v}_{1}\right)\backslash \{v,w\}|+|{N}_{{B}_{n}}\left({v}_{2}\right)\backslash \{v,w\}|=(n-1-2)+(n-1-2)+(n-1-2)+(n-1-2)=4n-12$. It follows that $|{F}_{2}|=|{F}_{2}\backslash {F}_{1}|+|{F}_{1}\cap {F}_{2}|\ge 1+4n-12=4n-11>2n-3\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}(n\ge 5)$, which contradicts $|{F}_{2}|\le 2n-3$. The proof of Claim 1 is complete.

Let $u\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$. 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 Lemma 10, by the condition (1) of Lemma 10, for any pair of adjacent vertices $u,w\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$, there is no vertex $v\in {F}_{1}\Delta {F}_{2}$ such that $uw\in E\left({B}_{n}\right)$ and $vw\in E\left({B}_{n}\right)$. It follows that u has no neighbor in ${F}_{1}\Delta {F}_{2}$. By the arbitrariness of u, there is no edge between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. Since ${F}_{2}\backslash {F}_{1}\ne \varnothing $ and ${F}_{1}$ is a 1-good-neighbor faulty set, ${\delta}_{{B}_{n}}\left([{F}_{2}\backslash {F}_{1}]\right)\ge 1$ and hence $|{F}_{2}\backslash {F}_{1}|\ge 2$. Since both ${F}_{1}$ and ${F}_{2}$ are 1-good-neighbor faulty sets, and there is no edge between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$, ${F}_{1}\cap {F}_{2}$ is a 1-good-neighbor cut of ${B}_{n}$. By Theorem 3, $|{F}_{1}\cap {F}_{2}|\ge 2n-4$. Therefore, $|{F}_{2}|=|{F}_{2}\backslash {F}_{1}|+|{F}_{1}\cap {F}_{2}|\ge 2+2n-4=2n-2$, which contradicts with that $|{F}_{2}|\le 2n-3$. So ${B}_{n}$ is 1-good-neighbor $(2n-3)$-diagnosable. By the definition of ${t}_{1}\left({B}_{n}\right)$, ${t}_{1}\left({B}_{n}\right)\ge 3n-4$. □

Combining Lemmas 13 and 14, we have the following theorem.

**Theorem**

**11.**

Let $n\ge 5$. Then the 1-good-neighbor diagnosability of the bubble-sort graph ${B}_{n}$ under the MM${}^{\ast}$ model is $2n-3$.

**Lemma**

**15.**

Let $n\ge 4$. Then the 2-good-neighbor diagnosability ${t}_{2}\left({B}_{n}\right)\le 4n-9$ under the MM${}^{\ast}$ model.

**Proof.**

Let A, ${F}_{1}$ and ${F}_{2}$ be defined in Lemma 2. By the Lemma 2, ${F}_{1}={N}_{{B}_{n}}\left(A\right)$, ${F}_{2}=A\cup {N}_{{B}_{n}}\left(A\right)$, then $|{F}_{1}|=4n-12$, $|{F}_{2}|=4n-8$, $\delta ({B}_{n}-{F}_{1})\ge 2$, and $\delta ({B}_{n}-{F}_{2})\ge 2$. So both ${F}_{1}$ and ${F}_{2}$ are 2-good-neighbor faulty sets. By the definitions of ${F}_{1}$ and ${F}_{2}$, ${F}_{1}\Delta {F}_{2}=A$. Note ${F}_{1}\backslash {F}_{2}=\varnothing $, ${F}_{2}\backslash {F}_{1}=A$ and $(V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2}))\cap A=\varnothing $. Therefore, 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, ${t}_{2}\left({B}_{n}\right)\le 4n-9$. The proof is complete. □

**Lemma**

**16.**

Let $n\ge 4$. Then the 2-good-neighbor diagnosability ${t}_{2}\left({B}_{n}\right)\ge 4n-9$ under the MM${}^{\ast}$ 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 $|{F}_{1}|\le 4n-9$ and $|{F}_{2}|\le 4n-9$, 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 ${F}_{2}\backslash {F}_{1}\ne \varnothing .$ Similarly to the discussion on $V\left({B}_{n}\right)\ne {F}_{1}\cup {F}_{2}$ in Lemma 5, we have $V\left({B}_{n}\right)\ne {F}_{1}\cup {F}_{2}$.

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 2-good neighbor faulty set, there are two vertices $u,v\in {F}_{2}\backslash {F}_{1}$ 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, $B{S}_{n}-{F}_{1}-{F}_{2}$ has no isolated vertex. The proof of Claim 1 is complete.

Let $u\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$. 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 $u,w\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$, there is no vertex $v\in {F}_{1}\Delta {F}_{2}$ such that $uw\in E\left({B}_{n}\right)$ and $vw\in E\left({B}_{n}\right)$. It follows that u has no neighbor in ${F}_{1}\Delta {F}_{2}$. By the arbitrariness of u, there is no edge between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. Since ${F}_{2}\backslash {F}_{1}\ne \varnothing $ and ${F}_{1}$ is a 2-good-neighbor faulty set, ${\delta}_{{B}_{n}}\left([{F}_{2}\backslash {F}_{1}]\right)\ge 2$. By Lemma 4, $|{F}_{2}\backslash {F}_{1}|\ge 4$. Since both ${F}_{1}$ and ${F}_{2}$ are 2-good-neighbor faulty sets, and there is no edge between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$, ${F}_{1}\cap {F}_{2}$ is a 2-good-neighbor cut of ${B}_{n}$. By Theorem 4, we have $|{F}_{1}\cap {F}_{2}|\ge 4n-12$. Therefore, $|{F}_{2}|=|{F}_{2}\backslash {F}_{1}|+|{F}_{1}\cap {F}_{2}|\ge 4+(4n-12)=4n-8$, which contradicts $|{F}_{2}|\le 4n-9$. Therefore, ${B}_{n}$ is 2-good-neighbor $(4n-9)$-diagnosable and ${t}_{2}\left({B}_{n}\right)\ge 4n-9$. The proof is complete. □

Combining Lemmas 15 and 16, we have the following theorem.

**Theorem**

**12.**

Let $n\ge 4$. Then the 2-good-neighbor diagnosability of the bubble-sort star graph ${B}_{n}$ under the $M{M}^{\ast}$ model is $4n-9$.

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] ${B}_{3}$ is not 2-diagnosable.

**Lemma**

**17.**

Let $n\ge 7$. Then the 3-good-neighbor diagnosability ${t}_{3}\left({B}_{n}\right)\le 8n-25$ under the MM${}^{\ast}$ model.

**Proof.**

Let A, ${F}_{1}$ and ${F}_{2}$ be defined in Lemma 6. By the Lemma 6, ${F}_{1}={N}_{{B}_{n}}\left(A\right)$, ${F}_{2}=A\cup {N}_{{B}_{n}}\left(A\right)$, then $|{F}_{1}|=8n-32$, $|{F}_{2}|=8n-24$, $\delta ({B}_{n}-{F}_{1})\ge 3$, and $\delta ({B}_{n}-{F}_{2})\ge 3$. So both ${F}_{1}$ and ${F}_{2}$ are 3-good-neighbor faulty sets. By the definitions of ${F}_{1}$ and ${F}_{2}$, ${F}_{1}\Delta {F}_{2}=A$. Note ${F}_{1}\backslash {F}_{2}=\varnothing $, ${F}_{2}\backslash {F}_{1}=A$ and $(V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2}))\cap A=\varnothing $. Therefore, both ${F}_{1}$ and ${F}_{2}$ are not satisfied with any one condition in Lemma 10, and ${B}_{n}$ is not 3-good-neighbor $(8n-24)$-diagnosable. Hence, ${t}_{2}\left({B}_{n}\right)\le 8n-25$. The proof is complete. □

**Lemma**

**18.**

Let $n\ge 7$. Then the 3-good-neighbor diagnosability ${t}_{3}\left({B}_{n}\right)\ge 8n-25$ under the MM${}^{\ast}$ model.

**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 $|{F}_{1}|\le 8n-25$ and $|{F}_{2}|\le 8n-25$, 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 ${F}_{2}\backslash {F}_{1}\ne \varnothing .$ Similarly to the discussion on $V\left({B}_{n}\right)\ne {F}_{1}\cup {F}_{2}$ in Lemma 9, we have $V\left({B}_{n}\right)\ne {F}_{1}\cup {F}_{2}$.

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 $u,v\in {F}_{2}\backslash {F}_{1}$ 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, $B{S}_{n}-{F}_{1}-{F}_{2}$ has no isolated vertex. The proof of Claim 1 is complete.

Let $u\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$. 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 $u,w\in V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$, there is no vertex $v\in {F}_{1}\Delta {F}_{2}$ such that $uw\in E\left({B}_{n}\right)$ and $vw\in E\left({B}_{n}\right)$. It follows that u has no neighbor in ${F}_{1}\Delta {F}_{2}$. By the arbitrariness of u, there is no edge between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$. Since ${F}_{2}\backslash {F}_{1}\ne \varnothing $ and ${F}_{1}$ is a 3-good-neighbor faulty set, ${\delta}_{{B}_{n}}\left([{F}_{2}\backslash {F}_{1}]\right)\ge 3$. By Lemma 8, $|{F}_{2}\backslash {F}_{1}|\ge 8$. Since both ${F}_{1}$ and ${F}_{2}$ are 3-good-neighbor faulty sets, and there is no edge between $V\left({B}_{n}\right)\backslash ({F}_{1}\cup {F}_{2})$ and ${F}_{1}\Delta {F}_{2}$, ${F}_{1}\cap {F}_{2}$ is a 3-good-neighbor cut of ${B}_{n}$. By Theorem 5, we have $|{F}_{1}\cap {F}_{2}|\ge 8n-32$. Therefore, $|{F}_{2}|=|{F}_{2}\backslash {F}_{1}|+|{F}_{1}\cap {F}_{2}|\ge 8+(8n-32)=8n-24$, which contradicts $|{F}_{2}|\le 8n-25$. Therefore, ${B}_{n}$ is 3-good-neighbor $(8n-25)$-diagnosable and ${t}_{3}\left({B}_{n}\right)\ge 8n-25$. The proof is complete. □

Combining Lemmas 17 and 18, we have the following theorem.

**Theorem**

**13.**

Let $n\ge 7$. Then the 3-good-neighbor diagnosability of the bubble-sort graph ${B}_{n}$ under the $M{M}^{\ast}$ model is $8n-25$.

## 5. 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${}^{\ast}$ 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${}^{\ast}$ 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.

## Funding

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

## Conflicts of Interest

The authors declare no conflict of interest.

## References

- Preparata, F.P.; Metze, G.; Chien, R.T. On the connection assignment problem of diagnosable systems. IEEE Trans. Comput.
**1967**, 16, 848–854. [Google Scholar] [CrossRef] - Barsi, F.; Grandoni, F.; Maestrini, P. A Theory of Diagnosability of Digital Systems. IEEE Trans. Comput.
**1976**, 25, 585–593. [Google Scholar] [CrossRef] - Maeng, J.; Malek, M. A comparison connection assignment for self-diagnosis of multiprocessor systems. In Proceedings of the 11th International Symposium on Fault-Tolerant Computing, Portland, OR, USA, 24–26 June 1981; pp. 173–175. [Google Scholar]
- Dahbura, A.T.; Masson, G.M. An O(n
^{2.5}) Fault identification algorithm for diagnosable systems. IEEE Trans. Comput.**1984**, 33, 486–492. [Google Scholar] [CrossRef] - Peng, S.-L.; Lin, C.-K.; Tan, J.J.M.; Hsu, L.-H. The g-good-neighbor conditional diagnosability of hypercube under PMC model. Appl. Math. Comput.
**2012**, 218, 10406–10412. [Google Scholar] [CrossRef] - Chang, N.-W.; Hsieh, S.-Y. Structural properties and conditional diagnosability of star graphs by using the PMC Model. IEEE Trans. Parallel Distrib. Syst.
**2014**, 25, 3002–3011. [Google Scholar] [CrossRef] - Cheng, E.; Lipták, L. Fault resiliency of Cayley graphs generated by transpositions. Int. J. Found. Comput. Sci.
**2007**, 18, 1005–1022. [Google Scholar] [CrossRef] - Fan, J. Diagnosability of crossed cubes under the comparison diagnosis model. IEEE Trans. Parallel Distrib. Syst.
**2002**, 13, 1099–1104. [Google Scholar] - Hu, S.-C.; Yang, C.-B. Fault tolerance on star graphs. In Proceedings of the First Aizu International Symposium on Parallel Algorithms/Architecture Synthesis, Aizu-Wakamatsu, Japan, 15–17 March 1995; pp. 176–182. [Google Scholar]
- Lai, P.-L.; Tan, J.J.M.; Chang, C.-P.; Hsu, L.-H. Conditional diagnosability measures for large multiprocessor systems. IEEE Trans. Comput.
**2005**, 54, 165–175. [Google Scholar][Green Version] - Lin, C.-K.; Tan, J.J.M.; Hsu, L.-H.; Cheng, E.; Lipták, L. Conditional diagnosability of Cayley graphs generated by transposition trees under the comparison diagnosis Model. J. Interconnect. Netw.
**2008**, 9, 83–97. [Google Scholar] [CrossRef] - Ren, Y.; Wang, S. Diagnosability of bubble-sort graph networks under the comparison diagnosis model. In Proceedings of the 2015 International Conference on Computational Intelligence and Communication Networks, Jabalpur, India, 12–14 December 2015; pp. 823–826. [Google Scholar]
- Wang, M.; Guo, Y.; Wang, S. The 1-good-neighbor diagnosability of Cayley graphs generated by transposition trees under the PMC model and MM
^{*}model. Int. J. Comput. Math.**2017**, 94, 620–631. [Google Scholar] [CrossRef] - Wang, M.; Lin, Y.; Wang, S. The 1-good-neighbor connectivity and diagnosability of Cayley graphs generated by complete graphs. Discret. Appl. Math.
**2018**, 246, 108–118. [Google Scholar] [CrossRef] - Wang, M.; Lin, Y.; Wang, S. The connectivity and nature diagnosability of expanded k-ary n-cubes. RAIRO-Theor. Inform. Appl.
**2017**, 51, 71–89. [Google Scholar] [CrossRef] - Wang, S.; Ren, Y. g-Good-neighbor diagnosability of arrangement graphs under the PMC model and MM* model. Information
**2018**, 9, 275. [Google Scholar] [CrossRef] - Wang, S.; Zhao, L. A note on the nature diagnosability of alternating group graphs under the PMC model and MM* model. J. Interconnect. Netw.
**2018**, 18, 1850005. [Google Scholar] [CrossRef] - Wang, S.; Wang, M. The g-good-neighbor and g-extra diagnosability of networks. Theor. Comput. Sci.
**2018**, in press. [Google Scholar] [CrossRef] - Wang, S.; Han, W. The g-good-neighbor conditional diagnosability of n-dimensional hypercubes under the MM* model. Inf. Process. Lett.
**2016**, 116, 574–577. [Google Scholar] [CrossRef] - Yuan, J.; Liu, A.; Ma, X.; Liu, X.; Qin, X.; Zhang, J. The g-good-neighbor conditional diagnosability of k-ary n-cubes under the PMC model and MM
^{*}model. IEEE Trans. Parallel Distrib. Syst.**2015**, 26, 1165–1177. [Google Scholar] [CrossRef] - Zheng, J.; Latifi, S.; Regentova, E.; Luo, K.; Wu, X. Diagnosability of star graphs under the comparison diagnosis Model. Inf. Process. Lett.
**2005**, 93, 29–36. [Google Scholar] [CrossRef] - Bondy, J.A.; Murty, U.S.R. Graph Theory; Springer: New York, NY, USA, 2007. [Google Scholar]
- Hungerford, T.W. Algebra; Springer: New York, NY, USA, 1974. [Google Scholar]
- Akers, S.B. Balakrishnan Krishnamurthy, A group-theoretic model for symmetric interconnection networks. IEEE Trans. Comput.
**1989**, 38, 555–566. [Google Scholar] [CrossRef] - Cheng, E.; Lipták, L.; Shawash, N. Orienting Cayley graphs generated by transposition trees. Comput. Math. Appl.
**2008**, 55, 2662–2672. [Google Scholar] [CrossRef][Green Version] - Yang, W.; Li, H.; Meng, J. Conditional connectivity of Cayley graphs generated by transposition trees. Inf. Process. Lett.
**2010**, 110, 1027–1030. [Google Scholar] [CrossRef] - Shi, L.-S.; Wu, P. Conditional connectivity of bubble sort graphs. Acta Math. Appl. Sin.
**2017**, 33, 933–944. [Google Scholar] [CrossRef]

**Figure 2.**Illustration of a distinguishable pair $({F}_{1},{F}_{2})$ under Preparata, Metze, and Chien’s (PMC) model.

**Figure 3.**Illustration of a distinguishable pair $({F}_{1},{F}_{2})$ under Maeng and Malek’s (MM)* model.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).