Equal relation between the extra connectivity and pessimistic diagnosability for some regular graphs

TL;DR

This paper establishes a fundamental equality between extra connectivity and pessimistic diagnosability in certain regular graphs, providing exact values for various well-known network topologies.

Contribution

It proves that under specific conditions, the pessimistic diagnosability equals the extra connectivity in regular graphs and computes these values for several important network models.

Findings

Proves $t_p(G) = \kappa_{1}(G)$ for some regular graphs.

Calculates exact diagnosability for split-star, Cayley, and burnt pancake networks.

Derives known results for various classical network topologies.

Abstract

Extra connectivity and the pessimistic diagnosis are two crucial subjects for a multiprocessor system's ability to tolerate and diagnose faulty processor. The pessimistic diagnosis strategy is a classic strategy based on the PMC model in which isolates all faulty vertices within a set containing at most one fault-free vertex. In this paper, the result that the pessimistic diagnosability $t_p(G)$ equals the extra connectivity $\kappa_{1}(G)$ of a regular graph $G$ under some conditions are shown. Furthermore, the following new results are gotten: the pessimistic diagnosability $t_p(S_n^2)=4n-9$ for split-star networks $S_n^2$, $t_p(\Gamma_n)=2n-4$ for Cayley graphs generated by transposition trees $\Gamma_n$, $t_p(\Gamma_{n}(\Delta))=4n-11$ for Cayley graph generated by the $2$-tree $\Gamma_{n}(\Delta)$, $t_{p}(BP_n)=2n-2$ for the burnt pancake networks $BP_n$. As corollaries, the known results about the extra connectivity and the pessimistic diagnosability of many famous networks including the alternating group graphs, the alternating group networks, BC networks, the $k$-ary $n$-cube networks etc. are obtained directly.

Equations

$$|N(x)\cap N(y)|=\left\{\begin{array}[]{ll}1&\ \text{if $xy\in E(Q_{n}^{k})$ and $k=3$};\\ 2&\ \text{if $xy\notin E(Q_{n}^{k})$ and $N(x)\cap N(y)\neq\emptyset$};\\ 0&\ \text{otherwise.}\end{array}\right.$$

$$|N(x)\cap N(y)|=\left\{\begin{array}[]{ll}1&\ \text{if $d(x,y)=1$};\\ 2&\ \text{if $d(x,y)=2$};\\ 0&\ \text{if $d(x,y)\geq 3$.}\end{array}\right.$$

Full text

1 Introduction

It is well known that a topological structure of an interconnection network can be modeled by a loopless undirected graph $G=(V,E)$, where vertices in $V$ represent the processors and the edges in $E$ represent the communication links. In this paper, we use graphs and networks interchangeably. The connectivity $\kappa(G)$ of a connected graph $G$ is the minimum number of vertices removed to get the graph disconnected or trivial. In a multiprocessor system, some processors may fail, connectivity is used to determine the reliability and fault tolerance of a network. However, connectivity is not suitable for large-scale processing systems because it is almost impossible for all processors adjacent to, or all links incident to, the same processors to fail simultaneously. To compensate for this shortcoming, it seems reasonable to generalize the notion of classical connectivity by imposing some conditions or restrictions on the components of $G$ when we delete the set of faulty processors. J. Fábrega and M.A. Fiol [16] introduced the extra connectivity of interconnection networks as follows.

Definition 1**.**

A vertex set $S\subseteq V(G)$ is called to be an $h$-extra vertex cut if $G-S$ is disconnected and every component of $G-S$ has at least $h+1$ vertices. The $h$-extra connectivity of $G$, denoted by $\kappa_{h}(G)$, is defined as the cardinality of a minimum $h$-extra vertex cut, if exists.

It is obvious that $\kappa_{0}(G)=\kappa(G)$ for any graph $G$ that is not a complete graph. The $1$-extra connectivity is usually called extra connectivity. The problem of determining the extra connectivity of numerous networks has received a great deal of attention in recent years. Interested readers may refer to [1, 20, 21, 25, 32, 33] or others for further details.

The diagnosis of a system is the process of appraising the faulty processors. A number of models have been proposed for diagnosing faulty processors in a network. Preparata et al. [34] first introduced a graph theoretical model, the so-called PMC model (i.e., Preparata, Metze and Chien s model), for system level diagnosis in multiprocessor systems. The pessimistic diagnosis strategy proposed by Kavianpour and Friedman [30] is a classic diagnostic model based on the PMC model. In this strategy, all faulty processors to be isolated within a set having at most one fault-free processor.

Definition 2**.**

A system is $t/t$-diagnosable, provided the number of faulty processors is bounded by $t$, all faulty processors can be isolated within a set of size at most $t$ with at most one fault-free vertex mistaken as a faulty one. The pessimistic diagnosability of a system $G$, denoted by $t_{p}(G)$, is the maximal number of faulty processors so that the system $G$ is $t/t$-diagnosable.

The pessimistic diagnosability of many interconnection networks has been explored. For example, see [17, 19, 30, 37, 38, 41] etc.

Based on the important of the extra connectivity and the pessimistic diagnosability and motivated by the recent researches on the extra connectivity and pessimistic diagnosability of some graphs, including some famous networks, our object is to propose the relationship between extra connectivity and pessimistic diagnosability of regular graphs with some given conditions. In this paper, the result that the pessimistic diagnosability $t_{p}(G)$ equals the extra connectivity $\kappa_{1}(G)$ of a regular graph $G$ under some conditions are shown. Furthermore, the following new results are gotten: the pessimistic diagnosability $t_{p}(S_{n}^{2})=4n-9$ for split-star networks $S_{n}^{2}$; $t_{p}(\Gamma_{n})=2n-4$ for Cayley graphs generated by transposition trees $\Gamma_{n}$; $t_{p}(\Gamma_{n}(\Delta))=4n-11$ for Cayley graph generated by the $2$-tree $\Gamma_{n}(\Delta)$; $t_{p}(BP_{n})=2n-2$ for the burnt pancake networks $BP_{n}$. As corollaries, the known results about the extra connectivity and the pessimistic diagnosability of many famous networks including the alternating group graphs, the alternating group networks, BC networks and the $k$-ary $n$-cube networks etc. are obtained directly.

The remainder of this paper is organized as follows. Section $2$ introduces necessary definitions and properties of some graphs. In Section 3, we determines the equal relationship between extra connectivity and pessimistic diagnosability of regular graphs with some given conditions. In Section 4, we concentrates on the applications to some famous networks. The pessimistic diagnosability and the extra connectivity of many famous networks, such as the alternating group graph $AG_{n}$, the alternating group network $AN_{n}$, the $k$-ary $n$-cube networks $Q_{n}^{k}$, the BC networks $X_{n}$, the split-star networks $S_{n}^{2}$, the Cayley graphs generated by transposition trees $\Gamma_{n}$, the Cayley graphs generated by $2$-trees $\Gamma_{n}(\Delta)$ and the burnt pancake networks $BP_{n}$ are obtained directly. Finally, our conclusions are given in Section 5.

2 Preliminaries

In this section, we give some terminologies and notations of combinatorial network theory. For notations not defined here, the reader is referred to [2].

We use a graph, denoted by $G=(V(G),E(G))$, to represent an interconnection network, where $V(G)$ is the vertex set of $G$; $E(G)$ is the edge set of $G$. For a vertex $u\in V(G)$, let $N_{G}(u)$ (or $N(u)$ if there is no ambiguity) denote a set of vertices in $G$ adjacent to $u$. For a vertex set $U\subseteq V(G)$, let $N_{G}(U)=\bigcup\limits_{v\in U}N_{G}(v)-U$ and $G[U]$ be the subgraph of $G$ induced by $U$. If $|N_{G}(u)|=k$ for any vertex in $G$, then $G$ is $k$-regular. For any two vertices $u$ and $v$ in $G$, let $cn(G;u,v)$ denote the number of vertices who are the neighbors of both $u$ and $v$, that is, $cn(G;u,v)=|N_{G}(u)\cap N_{G}(v)|$. Let $cn(G)=\max\{cn(G;u,v):u,v\in V(G)\}$, $l(G)=\max\{cn(G;u,v):(u,v)\in E(G)\}$. Let $|V(G)|$ be the size of vertex set and $|E(G)|$ be the size of edge set. Throughout this paper, all graphs are finite, undirected without loops.

Let $[n]=\{1,2,\ldots,n\}$ and $\langle n\rangle=\{-1,-2,\ldots,-n,1,2,\ldots,n\}$. For a finite group $A$ and a subset $S$ of $A$ such that $1\notin S$ and $S=S^{-1}$ (where $1$ is the identity element of $A$), the Cayley graph $\hbox{\rm Cay}(A;S)$ on $A$ with respect to $S$ is defined to have vertex set $A$ and edge set $\{(g,gs)|g\in A,s\in S\}$. A Cayley graph is $|S|$-regular, and is connected if and only if $S$ generates $\Gamma$. Moreover, A Cayley graph is $|S|$-connected if $S$ is a minimal generating set of $\Gamma$.

2.1 The alternating group graphs

Jwo et al. [29] introduced the alternating group graph as an interconnection network topology for computing systems.

Definition 3**.**

Let $A_{n}$ be the alternating group of degree $n$ with $n\geq 3$. Set $S=\{(1\ 2\ i),(1\ i\ 2)\ |\ 3\leq i\leq n\}$. The alternating group graph, denoted by $AG_{n}$, is defined as the Cayley graph $AG_{n}=\hbox{\rm Cay}(A_{n},S)$.

It is clear that $AG_{3}$ is a triangle, $AG_{n}$ is a $(2n-4)$-connected and $(2n-4)$-regular graph with $n!/2$ vertices. Each $AG_{n}$ contains $n$ sub-alternating group graphs $AG_{n}^{0},AG_{n}^{1},\ldots,AG_{n}^{n-1}$. For each $i\in[n]$, $AG_{n}^{i}$ is isomorphic to $AG_{n-1}$. For each vertex $v\in AG_{n}^{i}$, $v$ has exactly two neighbors that are not contained in $AG_{n}^{i}$, which are called the extra neighbors of $v$.

Lemma 1**.**

([24])* The extra neighbors of every vertex of $AG_{n}$ are in different subgraphs $AG_{n}^{i}$ for $n\geq 4$. For any two different vertices $u,v$, $cn(AG_{n}:u,v)=1$ if $u$ and $v$ are adjacent; otherwise, $cn(AG_{n}:u,v)\leq 2$.*

Lemma 2**.**

([37])* Let $AG_{n}$ be the $n$-dimensional alternating group graph for $n\geq 4$. If $U$ is a subset of $V(AG_{n})$ and $2\leq|U|\leq 8n-25$, then $|N_{AG_{n}}(U)|\geq 4n-11$.*

Lemma 3**.**

([24])* Let $F$ be a vertex-cut of $AG_{n}$ for $n\geq 5$. If $|F|\leq 4n-11$, then $AG_{n}-F$ satisfies one of the following conditions:*

1. (1)

$AG_{n}-F$* has two components, one of which is a trivial component.* 2. (2)

$AG_{n}-F$* has two components, one of which is an edge. Moreover, if $|F|=4n-11$, $F$ is formed by the neighbor of the edge.*

2.2 The alternating group networks

The alternating group network $AN_{n}$ was first proposed by Y. Ji [28] to improve upon the alternating group graph $AG_{n}$, studied by Jwo and others [29].

Definition 4**.**

([28])* Let $A_{n}$ be an alternating group of degree $n\geq 3$ and let $S=\{(1\ 2\ 3),(1\ 3\ 2),$ $(1\ 2)(3\ i)$ $|\ 4\leq i\leq n\}$. The alternating group network, denoted by $AN_{n}$, is defined as the Cayley graph $\hbox{\rm Cay}(A_{n},S)$.*

By the definition, we can get some properties about $AN_{n}$ [28]. $AN_{n}$ is a regular graph with $n!/2$ vertices and $n!(n-1)/4$ edges. $AN_{3}$ is a triangle. $AN_{4}$ contains four copies of $AN_{3}$. $AN_{n}$ contains $n$ copies of $AN_{n-1}$, say $AN_{n}^{0},AN_{n}^{1},\ldots,AN_{n}^{n-1}$. For each $i\in[n]$, $AN_{n}^{i}$ is isomorphic to $AN_{n-1}$. By Theorem 1 in [44], $AN_{n}$ is $(n-1)$-regular and $(n-1)$-connected.

Lemma 4**.**

([23])* Let $AN_{n}$ be the alternating group network for $n\geq 3$ .*

1. (1)

Each vertex in $AN_{n}$ has exactly one extra neighbor. 2. (2)

$AN_{n}$* has no $4$-cycle and $5$-cycle.* 3. (3)

Let $u$ and $v$ be any two distinct vertices of $AN_{n}$, then $cn(AN_{n}:u,v)\leq 1$.

Lemma 5**.**

([45])* Let $F$ be a vertex-cut of $AN_{n}$ for $n\geq 5$. If $|F|\leq 2n-5$, then $AN_{n}-F$ satisfies one of the following conditions:*

1. (1)

$AN_{n}-F$* has two components, one of which is a trivial component.* 2. (2)

$AN_{n}-F$* has two components, one of which is an edge. Moreover, if $|F|=2n-5$, $F$ is formed by the neighbor of the edge.*

2.3 BC networks

Definition 5**.**

The $1$-dimensional BC network $X_{1}$ is a complete graph with two vertices. The $n$-dimensional BC network $X_{n}$ is defined as follows: $V(X_{n})=V(G_{1})\cup V(G_{2})$ and $E(X_{n})=E(G_{1})\cup E(G_{2})\cup M$, where $G_{1},G_{2}\in L_{n-1}$, and $M$ is a perfect matching between $V(G_{1})$ and $V(G_{2})$, where $L_{k}=\{X_{k}:X_{k}\ {\rm is\ an}\ k$-${\rm dimensional\ BC\ network}\}$.

Lemma 6**.**

Let $G=X_{n}\in L_{n}$ for $n\geq 1$. Then

1. (1)

([18])* $|V(G)|=2^{n}$, $|E(G)|=n\cdot 2^{n-1}$, $G$ is $n$-regular and triangle-free.* 2. (2)

([18],[39])* $\kappa(G)=n$.* 3. (3)

([47])* $cn(G)=2$.*

Lemma 7**.**

([47])* For any $X_{n}\in L_{n}$, let $F\subseteq V(X_{n})$ with $|F|\leq 2n-3$ be a vertex-cut of $X_{n}$. Then $X_{n}-F$ has two components, one of which is a trivial component.*

2.4 The $k$-ary $n$-cube networks

Definition 6**.**

The $k$-ary $n$-cube, denoted by $Q_{n}^{k}$, where $k\geq 2$ and $n\geq 1$ are integers, is a graph consisting of $k^{n}$ vertices, each of these vertices has the form $u=u_{n-1}u_{n-2}\cdots u_{0}$, where $u_{i}\in\{0,1,\ldots,k-1\}$ for $0\leq i\leq n-1$. Two vertices $u=u_{n-1}u_{n-2}\cdots u_{0}$ and $v=v_{n-1}v_{n-2}\cdots v_{0}$ in $Q_{n}^{k}$ are adjacent if and only if there exists an integer $j$, where $0\leq j\leq n-1$, such that $u_{j}=v_{j}\pm 1($mod $k)$ and $u_{i}=v_{i}$ for every $i\in\{0,1,\ldots,n-1\}\setminus\{j\}$. In this case, $(u,v)$ is a j-dimensional edge.

For convenience, $``($mod $k)"$ does not appear in similar expressions in the remainder of the paper. Note that each vertex has degree $2n$ for $k\geq 3$ and has degree $n$ for $k=2$. Clearly, $Q_{1}^{k}$ is a cycle of length $k$, $Q_{n}^{2}$ is an $n$-dimensional hypercube, $Q_{2}^{k}$ is a $k\times k$ wrap-around mesh.

$Q_{n}^{k}$ can be partitioned over the $j$th-dimension, for a $j\in[n-1]$, into $k$ disjoint subcubes, denoted by $Q_{n-1}^{k}[0],Q_{n-1}^{k}[1],\ldots,Q_{n-1}^{k}[k-1]$, by deleting all the $j$-dimensional edges from $Q_{n}^{k}$. For convenience, abbreviate these as $Q[0],Q[1],\ldots,Q[k-1]$ if there is no ambiguity. Moreover, $Q[i]$ for $0\leq i\leq k-1$ is isomorphic to the $k$-ary $(n-1)$-cube. For each vertex $u\in V(Q[i])$, the neighbor which is not in $V(Q[i])$ is called the extra neighbor. For $i\in[k-1]$, $u\in V(Q[i])$, the two extra neighbors of $u$ are in different subgraphs $Q[i+1]$ and $Q[i-1]$, respectively.

Lemma 8**.**

Let $Q_{n}^{k}$ be a $k$-ary $n$-cube, where $k\geq 2$ and $n\geq 1$ are integers.

1. (1)

([14])* $Q_{n}^{k}$ is $2n$-regular and $2n$-connected for $k\geq 3$ and $n$-regular and $n$-connected for $k=2$.* 2. (2)

([13, 20, 25])* For any $x,y\in V(Q_{n}^{k})$, $k\geqslant 2$,*

[TABLE]

Lemma 9**.**

1. (1)

([15])* If $F\subseteq V(Q_{n}^{2})$ with $|F|\leq 2n-3$ is a vertex cut of $Q_{n}^{2}$ for $n\geq 2$, then $Q_{n}^{2}-F$ has two components, one of which is a trivial component.* 2. (2)

([13, 20])* If $F\subseteq V(Q_{n}^{3})$ with $|F|\leq 4n-4$ is a vertex cut of $Q_{n}^{3}$ for $n\geq 2$, then $Q_{n}^{3}-F$ has two components, one of which is a trivial component.* 3. (3)

([13, 21])* If $F\subseteq V(Q_{n}^{k})$ is a vertex cut of $Q_{n}^{k}$ with $|F|\leq 4n-3$ for $n\geq 2$ and $k\geq 4$, then $Q_{n}^{k}-F$ has two components, one of which is a trivial component.*

2.5 Split-star networks $S_{n}^{2}$

Cheng et al. [7] propose the Split-star networks as alternatives to the star graphs and companion graphs with the alternating group graphs.

Definition 7**.**

Given two positive integers $n$ and $k$ with $n>k$, note that $[n]=\{1,2,\ldots,n\}$, and let $\mathcal{P}_{n}$ be a set of $n!$ permutations on $[n]$. The $n$-dimensional Split-star network, denoted by $S_{n}^{2}$, such that $V(S_{n}^{2})=\mathcal{P}_{n}$, $E(S_{n}^{2})=\{(p,q)|\ p$ (resp. $q$) can be obtained from $q$ (resp. $p$) by either a $2$-exchange or a $3$-rotation $\}$. Where

1. (1)

A $2$-exchange interchanges the symbols in 1st position and 2nd position. 2. (2)

A $3$-rotation rotates the symbols in three positions labeled by the vertices of a triangle in which three vertices of the triangle are $1,2$ and $k$ for some $k\in\{3,4,\ldots,n\}$.

Let $V_{n}^{n:i}$ be the set of all vertices in $S_{n}^{2}$ with the $n$th position having value $i$, i.e., $V_{n}^{n:i}=\{p|p=x_{1}x_{2}\cdots x_{n-1}i$, $x_{j}\in\{1,2,\ldots,i-1,i+1,\ldots n\}$ ($1\leq j\leq n-1$) are do not care symbols $\}$. The set $\{V_{n}^{n:i}|1\leq i\leq n\}$ forms a partition $V(S_{n}^{2})$. Let $S_{n}^{2:i}$ denote the subgraph of $S_{n}^{2}$ induced by $V_{n}^{n:i}$, i.e., $S_{n}^{2:i}=S_{n}^{2}[V_{n}^{n:i}]$. It is easy to know that $S_{n}^{2:i}$ is isomorphic to $S_{n-1}^{2}$. Every vertex $v\in S_{n}^{2:i}$ has exactly two neighbors, called extra neighbors, outside of $S_{n}^{2:i}$; moreover these two neighbors belong to different $S_{n}^{2:j}$s where $j\neq i$. We call these neighbors as the extra neighbors of $v$. We call these edges, whose end-vertices belong to different subgraphs, as cross edges. Let $S_{n,E}^{2}$ be a subgraph of $S_{n}^{2}$ induced by the set of even permutations, in which the adjacency rule is precisely the $3$-rotation. We know that $S_{n,E}^{2}$ is the alternating group graph $AG_{n}$ [29]. Let $S_{n,O}^{2}$ be a subgraph of $S_{n}^{2}$ induced by the set of odd permutations, in which the adjacency rule is precisely the $3$-rotation. We have that $S_{n,O}^{2}$ is also isomorphic to $AG_{n}$ and $S_{n,O}^{2}$ is isomorphic $S_{n,E}^{2}$ via the $2$-exchange $\phi(a_{1}a_{2}a_{3}\cdots a_{n})=a_{2}a_{1}a_{3}\cdots a_{n}$. Hence, there are $\frac{n!}{2}$ matching edges between $S_{n,O}^{2}$ and $S_{n,E}^{2}$. Indeed, the Split-star network $S_{n}^{2}$ is introduced in [8] which is the companion graph of $AG_{n}$.

Lemma 10**.**

( [6, 7, 8])* Let $S_{n}^{2}$ be the $n$-dimensional split-star network.*

1. (1)

$S_{n}^{2}$* is $(2n-3)$-regular and $\kappa(S_{n}^{2})=2n-3$ for $n\geq 2$.* 2. (2)

Two extra neighbors of every vertex in $S_{n}^{2:i}$ are in distinct induced subgraphs and these two extra neighbors are adjacent. For any two vertices in the same subgraph $S_{n}^{2:i}$, their extra neighbors in other subgraphs are different. There is one to one correspondence between the subgraph $S_{n,O}^{2}$ and the subgraph $S_{n,E}^{2}$. 3. (3)

Let $x,y$ be any two vertices of $S_{n}^{2}$, then

[TABLE]

Lemma 11**.**

( [32])* If $F\subseteq V(S_{n}^{2})$ with $|F|\leq 4n-10$ is a vertex cut of $S_{n}^{2}$ for $n\geq 4$, then $S_{n}^{2}-F$ has two components, one of which is a trivial component.*

2.6 Cayley graphs generated by transposition trees $\Gamma_{n}$

Note that $\mathcal{P}_{n}$ is a group of all permutations on $[n]$. For convenience, $(ij)$, which is called a transposition, denotes the permutation that swaps the elements at position $i$ and $j$, that is $(ij)p_{1}p_{2}\ldots p_{i}\ldots p_{j}\ldots p_{n}=p_{1}p_{2}\ldots p_{j}\ldots p_{i}\ldots p_{n}$.

Definition 8**.**

Let $\mathcal{P}_{n}$ be symmetric group on $[n]$, and the generating set $S$ to be a set of transpositions. A graph $G(S)$ with vertex set $[n]$, where there is an edge between $i$ and $j$ if and only if the transposition $(ij)$ belongs to $S$, is called the transposition generating graph. When $G(S)$ is a tree, we call $G(S)$ a transposition tree. The Cayley graphs $Cay(\mathcal{P}_{n},S)$ obtained by transposition trees are called Cayley graphs generated by transposition trees, denoted by $\Gamma_{n}$.

If $G(S)\cong K_{1,n-1}$, $Cay(\mathcal{P}_{n},S)$ is called the star graph, denoted by $S_{n}$. If $G(S)\cong P_{n}$, that is the transposition tree is a path $P_{n}$ with $n$ vertices, then $Cay(\mathcal{P}_{n},S)$ is called the bubble-sort graph, denoted by $B_{n}$.

Let $\Gamma_{n}^{i}$ be the subgraph of $\Gamma_{n}$ spanned by vertices corresponding to permutations with $i$ in the last position. Then $\Gamma_{n}$ can be divided into $n$ subgraphs $\Gamma_{n-1}^{1}$, $\Gamma_{n-1}^{2}$, $\cdots$, $\Gamma_{n-1}^{n}$ and each $\Gamma_{n-1}^{i}$ is isomorphic to $\Gamma_{n-1}$ for $i\in[n]$. For $u\in V(\Gamma_{n-1}^{i})$, denoted by $u^{\prime}=u(1n)$ the unique neighbor of $u$ outside $\Gamma_{n-1}^{i}$, called the extra neighbor of $u$.

Lemma 12**.**

Let $\Gamma_{n}$ be the Cayley graphs generated by transposition trees for $n\geq 3$.

1. (1)

([4])* $\kappa(\Gamma_{n})=n-1$.* 2. (2)

([4])* $\Gamma_{n}$ has the girth $4$ unless $\Gamma_{n}$ is the star graph which has girth $6$. $\Gamma_{n}$ does not have $K_{2,3}$ as a subgraph.* 3. (3)

([42])* For any two distinct vertices $u,v\in\Gamma_{n}$, $|N_{\Gamma_{n}}(u)\cap N_{\Gamma_{n}}(v)|=1$ if $\Gamma_{n}=S_{n}$; Otherwise $|N_{\Gamma_{n}}(u)\cap N_{\Gamma_{n}}(v)|\leq 2$.*

Lemma 13**.**

([4, 42])* If $F\subseteq V(\Gamma_{n})$ with $|F|\leq 2n-5$ is a vertex cut of $\Gamma_{n}$ for $n\geq 4$, then $\Gamma_{n}-F$ has two components, one of which is a trivial component.*

2.7 Cayley graphs generated by $2$-trees

Definition 9**.**

Let $\Gamma$ be the alternating group, the set of even permutations on $\{1,2,\ldots,n\}$, and the generating set $\Delta$ to be a set of $3$-cycles. To get an undirected Cayley graph, we will assume that whenever a $3$-cycle $(abc)$ is in $\Delta$, so is its inverse, $(acb)$. Since $(abc)$, $(bca)$ and $(cab)$ represent the same permutation, the set $\{a,b,c\}$ uniquely represents this $3$-cycle and its inverse. So we can depict $\Delta$ via a hypergraph with vertex set $[n]$, where a hyperedge of size $3$ corresponds to each pair of a $3$-cycle and its inverse in $\Delta$.

It is easy to see that the Cayley graph generated by the $3$-cycles in $\Delta$ is connected if its corresponding hypergraph $H$ is connected. Since an interconnection network needs to be connected, we require $H$ graph to be connected.

In general, this graph may have extra $K_{3}$’s formed by vertices that do not correspond to a $3$-cycle in $\Delta$. We will avoid this possibility by considering a simpler case when $H$ has a tree-like structure. Such a graph is built by the following procedure. We start from $K_{3}$, then repeatedly add a new vertex, joining it to exactly two adjacent vertices of the previous graph. Any graph obtained by this procedure is called a $2$-tree. If $v$ is a vertex of a $2$-tree $H$ with the property that $H$ can be generated in such a way that $v$ is the last vertex added, then $v$ is called a leaf of the $2$-tree.

The alternating group graph $AG_{n}$ [28], can be viewed as the Cayley graph generated by the graph having a tree-like (in fact, star-like) structure of triangles.

It is easy to prove that if two $2$-trees are isomorphic, then the corresponding Cayley graphs will also be isomorphic; hence without loss of generality we may assume that vertex n is the tail of the $2$-tree. For $n\geq 4$, the vertices corresponding to even permutations ending with $i$ induce a subgraph $\Gamma_{n-1}^{i}(\Delta)$ that is also a Cayley graph generated by a 2-tree $\Delta^{\prime}$, which is obtained by deleting the edges corresponding to the two $3$-cycles in $\Delta$ containing $n$. Thus we obtain the following result of the recursive structure of $\Gamma_{n}(\Delta)$:

Lemma 14**.**

([9])* Let $\Gamma_{n}(\Delta)$ be a Cayley graph generated by the $2$-tree $\Delta$, $\Delta^{\prime}=\Delta-\{n\}$, $n\geq 4$. Then*

1. (1)

$\Gamma_{n}(\Delta)$* consists of $n$ vertex-disjoint subgraphs, $\Gamma_{n-1}^{1}(\Delta),\Gamma_{n-1}^{2}(\Delta),\ldots,\Gamma_{n-1}^{n}(\Delta)$, each isomorphic to $\Gamma_{n-1}(\Delta^{\prime})$.* 2. (2)

$\Gamma_{n-1}^{i}(\Delta)$* has $(n-1)!/2$ vertices, and it is $(2n-6)$-regular for all $i$.* 3. (3)

There are exactly $(n-2)!$ independent edges between $\Gamma_{n-1}^{i}(\Delta)$ and $\Gamma_{n-1}^{j}(\Delta)$ for all $i\neq j$. 4. (4)

Each vertex in $\Gamma_{n-1}^{i}(\Delta)$ has exactly two neighbors outside $\Gamma_{n-1}^{i}(\Delta)$; these two outside neighbors are in different $\Gamma_{n-1}^{k}(\Delta)$’s, and there is an edge between them. Thus every vertex forms a triangle with its two outside neighbors. 5. (5)

$\Gamma_{n}(\Delta)$* does not contain $K_{4}-e$, that is, $K_{4}$ with an edge deleted, and $K_{2,3}$ as a subgraph. For any two vertices $u$ and $v$, $|N(u)\cap N(v)|=1$ if $d(u,v)=1$, $|N(u)\cap N(v)|\leq 2$ otherwise.*

Lemma 15**.**

([3])* Let $G=\Gamma_{n}(\Delta)$ be a Cayley graph generated by the $2$-tree $\Delta$ for $n\geq 4$. Then $G$ is maximally connected, i.e., $G$ is $(2n-4)$-regular and $(2n-4)$-connected.*

Lemma 16**.**

([3])* Let $G=\Gamma_{n}(\Delta)$ be a Cayley graph generated by the $2$-tree $\Delta$ for $n\geq 4$, and let $T$ be a set of vertices in $G$ such that $|T|\leq 4n-11$. If $n\geq 5$, then $G-T$ satisfies one of the following conditions:*

1. (1)

$G-T$* is connected.* 2. (2)

$G-T$* has two components, one of which is a singleton.* 3. (3)

$G-T$* has two components, one of which is a $K_{2}$. Moreover, $|T|=4n-11$, and the set $T$ is formed by the neighbors of the two vertices in the $K_{2}$.*

When $n=4$, there are two additional possibilities. In both cases, $G-T$ has two components, one of which is a $4$-cycle. The other component is either a $4$-cycle if $|T|=4$ or a path with $3$ vertices if $|T|=5$.

2.8 Burnt pancake networks $BP_{n}$

Gates and Papadimitriou [22] introduced the burnt pancake problem in 1979. Burnt pancake problem relates to the construction of networks of parallel processors.

Let $n$ be a positive integer. We use $[n]$ to denote the set $\{1,2,\ldots,n\}$. To save space, the negative sign may be placed on the top of an expression. Thus, $\bar{i}=-i$. We use $\langle n\rangle$ to denote the set $[n]\cup\{\bar{i}|i\in[n]\}$. A signed permutation of $[n]$ is an $n$-permutation $u_{1}u_{2}\cdots u_{n}$ of $\langle n\rangle$ such that $|u_{1}||u_{2}|\cdots|u_{n}|$ taking the absolute value of each element, forms a permutation of $[n]$. For a signed permutation $u=x_{1}x_{2}\cdots x_{i}\cdots x_{n}$ of $\langle n\rangle$, the $i$-th prefix reversal of u, denoted by $u^{i}$ is $u^{i}=\bar{x}_{i}\bar{x}_{i-1}\cdots\bar{x}_{1}x_{i+1}\cdots x_{n},1\leq i\leq n$. For example, let $u=1\bar{2}4\bar{3}5$; then $u$ is a signed permutation of $[5]$, $u^{2}=2\bar{1}4\bar{3}5$, $u^{5}=\bar{5}3\bar{4}2\bar{1}$.

Definition 10**.**

An $n$-dimensional burnt pancake network $BP_{n}$ is defined to be an $n$-regular graph $G$ with $n!2^{n}$ vertices, each of which has a unique label from the signed permutation of $\langle n\rangle$. Two vertices $u$ and $v$ are adjacent in $BP_{n}$ if and only if $u^{i}=v$ for some unique $i$ ($1\leq i\leq n$). Such an edge $uv$ is called an $i$-dimensional edge and $v$ is called the $i$-neighbor of $u$. It is seen that every vertex has a unique $i$-neighbor for $1\leq i\leq n$.

Lemma 17**.**

([10, 12, 27])* An $n$-dimensional burnt pancake network $BP_{n}$ has the following combinatorial properties.*

1. (1)

$BP_{n}$* is $n$-regular with $n!\times 2^{n}$ vertices and $n!\times 2^{n-1}$ edges.* 2. (2)

$\kappa(BP_{n})=n$, the girth of $BP_{n}$($n\geq 3$) is $g(BP_{n})=8$. 3. (3)

$BP_{n}$* can be decomposed into $2n$ vertex-disjoint subgraphs, denoted $BP_{n}^{i}$, by fixing the symbol in the last position $n$, in which the symbol in the $n$th position is $i$, where $i\in[n]$. Obviously, $BP_{n}^{i}$ is isomorphic to $BP_{n-1}$. The number of cross edges between any two subgraphs, $BP_{n}^{i}$ and $BP_{n}^{j}$ ($i\neq j,i,j\in[n]$), is $|E(i,j)|=(n-2)!\times 2^{n-2}$ if $i\neq\bar{j}$; otherwise, $|E(i,j)|$=0. For a vertex $v\in V(BP_{n}^{i})$, $v$ has exactly one neighbor outside $BP_{n}^{i}$, called the extra neighbor of $v$.*

Lemma 18**.**

([35])* For any subset $F\subseteq V(BP_{n})$ with $|F|\leq 2n-2$ is a vertex-cut of $BP_{n}$ for $n\geq 4$, then $BP_{n}-F$ satisfies one of the following conditions.*

1. (1)

$BP_{n}-F$* has two connected components, one of which is a trivial component;* 2. (2)

$BP_{n}-F$* has two connected components, one of which is an edge. Furthermore, $F$ is the neighborhood of this edge with $|F|=2n-2$.*

3 Main result

In this section, the relationship between the pessimistic diagnosability under the PMC model and the extra connectivity with some restricted conditions will be proposed.

Lemma 19**.**

Let $G$ be a $k$-regular graph. Let $u$ and $v$ be two distinct vertices in $G$, if $cn(G;u,v)\leq 2$, then $|N_{G}(\{u,v\})|\geq 2k-2-l$, where $l=l(G)=\max\{cn(G;u,v):(u,v)\in E(G)\}$, i.e., $l=l(G)$ be the maximum number of common neighbors between any two adjacent vertices.

Proof.   Since $cn(G;u,v)\leq 2$, if $u$ is non-adjacent to $v$, then $|N_{G}(\{u,v\})|=|N_{G}(u)|+|N_{G}(v)|-cn(G;u,v)\geq 2k-2\geq 2k-2-l$. Otherwise, $u$ is adjacent to $v$, $|N_{G}(\{u,v\})|=|N_{G}(u)|-1+|N_{G}(v)|-1-cn(G;u,v)\geq 2(k-1)-l$. As a result, $|N_{G}(\{u,v\})|\geq 2k-2-l$. ∎

Tsai and Chen [36] derived the following result which characterizes a graph for $t/t$-diagnosability.

Lemma 20**.**

([36])* A graph $G$ is $t/t$-diagnosable if and only if for each vertex set $S\subseteq V(G)$ with $|S|=p$, $0\leq p\leq t-1$, $G-S$ has at most one trivial component and each nontrivial component $C$ of $G-S$ satisfies $|V(C)|\geq 2(t-p)+1$.*

The following result is useful.

Lemma 21**.**

([17])* Let $G$ be a connected graph and $U\subseteq V(G)$. Then, $|N_{V(G)-U}(U)|\geq\kappa(G)$ if $|V(G)-U|\geq\kappa(G)$, otherwise, $|N_{V(G)-U}(U)|=|V(G)-U|$.*

Theorem 1**.**

Let $G$ be a $k$-regular $k$-connected ($k\geq 5$) graph with order $N$. Let $U$ be a subset of $V(G)$ and $l=l(G)$ be the maximum number of common neighbors between any two adjacent vertices. Suppose further that all of the following conditions hold:

1. (1)

$N\geq 4k-2$. 2. (2)

$cn(G)\leq 2$. 3. (3)

If $2\leq|U|\leq 2(2k-4-l)$, then $|N_{G}(U)|\geq 2k-2-l$. 4. (4)

Let $F\subseteq V(G)$ be a vertex-cut of $G$. If $|F|\leq 2k-3-l$, then $G-F$ has a large component and a small component which is a trivial component.

Then, $t_{p}(G)=2k-2-l=\kappa_{1}(G)$.

Proof.   We first prove $t_{p}(G)\leq 2k-2-l$. Suppose $t_{p}(G)\geq 2k-2-l+1$, then $G$ is $(2k-2-l+1)/(2k-2-l+1)$-diagnosable. Let $(u,v)$ be an edge of $G$ such that $|N_{G}(u)\cap N_{G}(v)|=l$. Let $S=N_{G}(\{u,v\})$. Then $|S|=2k-2-l\leq t_{p}(G)-1$. An edge $\{u,v\}$ is a connected component of $G-S$, say $C$. By Lemma 20, $|V(C)|\geq 2(t_{p}(G)-|S|)+1\geq 2[(2k-2-l+1)-(2k-2-l)]+1=3$, which is a contradiction. Thus, $t_{p}(G)\leq 2k-2-l$.

Secondly, we show $t_{p}(G)\geq 2k-2-l$, i.e., $G$ is $(2k-2-l)/(2k-2-l)$-diagnosable. Suppose $G$ is not $(2k-2-l)/(2k-2-l)$-diagnosable, by Lemma 20, there exists a vertex set $S\subseteq V(G)$ with $|S|=p$, $0\leq p\leq 2k-3-l$ such that $G-S$ contains more than one trivial components or contains a nontrivial component $C$ with $|V(C)|\leq 2(2k-2-l-p)$. The following cases should be considered.

Case 1. $G-S$ contains more than one trivial components.

Suppose $C_{1}=\{u\}$ and $C_{2}=\{v\}$ are two distinct trivial components of $G-S$. By Condition (2) and Lemma 19, $|N_{G}(\{u,v\})|\geq 2k-2-l$. Note that $N_{G}(\{u,v\})\subseteq S$, this implies that $|S|\geq 2k-2-l$, which is a contradiction.

Case 2. $G-S$ contains a nontrivial component $C$ with $2\leq|V(C)|\leq 2(2k-2-l-p)$.

Suppose $p\leq 1$. Since the connectivity of $G$ is $k\geq 5>p$, $G-S$ is connected. It implies $C=G-S$. By $|V(C)|=|V(G)|-|S|=N-p\geq N-1$, Condition $(1)$ and $l\leq cn(G)\leq 2$, one has $|V(C)|\geq 4k-3\geq 2(2k-2-l-p)+1$ which is a contradiction.

Now consider $2\leq p\leq 2k-3-l$. Since $2\leq|V(C)|\leq 2(2k-2-l-p)$, so $2\leq|V(C)|\leq 2(2k-4-l)$. By condition $(3)$, $|N_{G}(V(C))|\geq 2k-2-l$. Since $C$ is a connected component of $G-S$, $N_{G}(V(C))\subseteq S$. This implies $p=|S|\geq 2k-2-l$, which is a contradiction for the fact that $p=|S|\leq 2k-3-l$. Thus, $t_{p}(G)\leq 2k-2-l$.

Next we prove $2k-2-l=\kappa_{1}(G)$. Let $(u,v)$ be an edge of $G$ such that $|N_{G}(u)\cap N_{G}(v)|=l$. Let $S=N_{G}(\{u,v\})$. Then $|S|=2k-2-l$. If $G-S=\{(u,v)\}$, then $|V(G)|=|S|+2=2k-l<4k-2$ for $k\geq 5$ which contradicts with Condition $(1)$. If $G-S$ has a trivial component which contains only one vertex, say $\{x\}$, then $G-S$ has at least two components: $\{x\}$ and the edge $(u,v)$. By $cn(G)\leq 2$, then $|S|\geq 2k-2-l+(k-4)=3k-6-l$. Note $3k-6-l>2k-2-l$ for $k\geq 5$, it is a contradiction. Thus, $G-S$ has no trivial component, i.e., $S$ is an extra vertex cut of $G$, which implies $\kappa_{1}(G)\leq 2k-2-l$. On the other hand, by condition $(4)$, $\kappa_{1}(G)\geq 2k-2-l$. Thus, $\kappa_{1}(G)=2k-2-l$.

By above discussion, $t_{p}(G)=2k-2-l=\kappa_{1}(G)$. ∎

4 Application to some interconnection networks

As applications of Theorem 1, in this section, we determine the pessimistic diagnosability and extra connectivity for some well-known interconnection networks, including the alternating group graph $AG_{n}$, the alternating group network $AN_{n}$, the $k$-ary $n$-cube networks $Q_{n}^{k}$, BC networks $X_{n}$, split-star networks $S_{n}^{2}$, Cayley graphs generated by transposition trees $\Gamma_{n}$, Cayley graphs generated by $2$-trees, burnt pancake networks $BP_{n}$.

4.1 Application to the alternating group graphs $AG_{n}$

Remark 1**.**

It is known that $\kappa_{1}(AG_{n})=4n-11$ for $n\geq 5$ determined by Lin et al. [33] and $t_{p}(AG_{n})=4n-11$ obtained by Tsai [37]. As a corollary of Theorem 1, we immediately obtain the following result which contains the above result.

Corollary 1**.**

Let $AG_{n}$ be the $n$-dimensional alternating group graph for $n\geq 5$. Then $t_{p}(AG_{n})=4n-11=\kappa_{1}(AG_{n})$.

Proof.   Obviously, $N=|V(AG_{n})|=\frac{n!}{2}$, $k=2n-4\geq 6$ for $n\geq 5$, $l=l(AG_{n})=1$.

Note that $N=\frac{n!}{2}\geq 4(2n-4)-2$ for $n\geq 5$, Conditions $(1)$ in Theorem 1 holds. Conditions $(2)-(4)$ in Theorem 1 hold by Lemmas 1, 2 and 3, respectively. Thus, $AG_{n}$ satisfies all conditions in Theorem 1, $t_{p}(AG_{n})=4n-11=\kappa_{1}(AG_{n})$ for $n\geq 5$. ∎

4.2 Application to the alternating group networks

Zhou [45] derived $\kappa_{1}(AN_{n})=2n-5$ for $n\geq 4$. However, $t_{p}(AN_{n})$ has not been determined so far. We can deduce the result as a corollary of Theorem 1 as following. Notice that for $AN_{n}$, $k=n-1$, $l=1$ in Theorem 1.

Lemma 22**.**

Let $AN_{n}$ be the $n$-dimensional alternating group network for $n\geq 4$. If $U$ is a subset of $V(AN_{n})$ and $2\leq|U|\leq 2(2k-4-l)=4n-14$, then $|N_{AN_{n}}(U)|\geq 2n-5$.

Proof.   The Lemma can be proved by using the induction on $n$. It is easy to verify that $|N_{AN_{4}}(U)|\geq 3$ for $|U|=2$ by Lemma 19. We assume that the lemma is true for $AN_{m}$, where $m$ is an integer with $5\leq m\leq n-1$, we will prove the result for $AN_{n}$.

Recall that $AN_{n}$ is constructed by $n$ disjoint $AN_{n-1}$’s, denoted by $AN_{n}^{i}$ for $i\in[n]$. Let $U_{i}=U\cap V(AN_{n}^{i})$ and $\overline{AN_{n}^{i}}=AN_{n}-AN_{n}^{i}$ for $i\in[n]$. Without loss of generality, we may assume that $|U_{1}|\geq|U_{2}|\geq\ldots\geq|U_{n}|$. The following cases should be considered.

Case 1. $|U_{1}|\leq 1$.

In this case, $|U_{i}|\leq 1$ for all $i\in[n]$. Clearly, $2\leq|U|\leq n$ because of $i\leq n$. The Lemma follows if $|U|=2$ by Lemma 19. Now assume that $3\leq|U|\leq n$. Since $AN_{n}$ is $(n-1)$-regular and $AN_{n}^{i}$ is isomorphic to $AN_{n-1}$, $|N_{AN_{n}}(U)|\geq 3\kappa(AN_{n}^{i})=3(n-2)\geq 2n-5$ for $n\geq 7$.

Case 2. $2\leq|U_{1}|\leq 4n-19$.

By inductive hypothesis in $AN_{n}^{1}$, $|N_{AN_{n}^{1}}(U_{1})|\geq 2(n-1)-5=2n-7$. If $U=U_{1}$, $|N_{AN_{n}}(U)|=|N_{AN_{n}^{1}}(U_{1})|+|N_{\overline{AN_{n}^{1}}}(U_{1})|\geq 2n-7+|U_{1}|\geq 2n-5$. Assume $U\neq U_{1}$ in the following. If $|U_{2}|=1$, $|N_{AN_{n}^{2}}(U_{2})|=\kappa(AN_{n}^{2})=n-2$. Note that $AN_{n}^{1}$ and $AN_{n}^{2}$ are vertex disjoint, $|N_{AN_{n}}(U)|\geq|N_{AN_{n}^{1}}(U_{1})|+|N_{AN_{n}^{2}}(U_{2})|\geq 3n-9\geq 2n-5$ for $n\geq 5$. Now consider $2\leq|U_{2}|\leq|U_{1}|\leq 4n-19$, by inductive hypothesis in $AN_{n}^{2}$, $|N_{AN_{n}^{2}}(U_{2})|\geq 2(n-1)-5=2n-7$. Thus, $|N_{AN_{n}}(U)|\geq|N_{AN_{n}^{1}}(U_{1})|+|N_{AN_{n}^{2}}(U_{2})|\geq 4n-14\geq 2n-5$ for $n\geq 5$.

Case 3. $4n-18\leq|U_{1}|\leq 4n-14$.

Since the connectivity of $AN_{n}^{1}$ is $n-2$, and $\frac{(n-1)!}{2}-|U_{1}|\geq n-2=\kappa(AN_{n}^{1})$ for $n\geq 5$, by Lemma 21, $|N_{AN_{n}^{1}}(U_{1})|\geq n-2$. By Lemma 4, $|N_{\overline{AN_{n}^{1}}}(U_{1})|=|U_{1}|$. If $U=U_{1}$, $|N_{AN_{n}}(U)|\geq|N_{AN_{n}^{1}}(U_{1})|+|N_{\overline{AN_{n}^{1}}}(U_{1})|\geq(n-2)+4n-18=5n-20\geq 2n-5$ for $n\geq 5$. In the following, we assume the case of $U\neq U_{1}$. Note that $U\neq U_{1}$ and $|U-U_{1}|\leq 3$, so $1\leq|U_{2}|\leq 3$.

If $|U_{2}|=1$, recall that $AN_{n}$ is $(n-1)$-regular and $AN_{n}^{i}$ is isomorphic to $AN_{n-1}$, $|N_{AN_{n}^{2}}(U_{2})|=\kappa(AN_{n}^{2})=n-2$. Hence, $|N_{AN_{n}}(U)|\geq|N_{AN_{n}^{1}}(U_{1})|+|N_{AN_{n}^{2}}(U_{2})|\geq 2n-4\geq 2n-5$ for $n\geq 5$. Now suppose that $2\leq|U_{2}|\leq 3$. Since $\frac{(n-1)!}{2}-|U_{2}|\geq n-2=\kappa(AN_{n}^{2})$ for $n\geq 5$, by Lemma 21, $|N_{AN_{n}^{2}}(U_{2})|\geq n-2$. Thus, $|N_{AN_{n}}(U)|\geq|N_{AN_{n}^{1}}(U_{1})|+|N_{AN_{n}^{2}}(U_{2})|\geq 2(n-2)\geq 2n-5$ for $n\geq 5$.

By the above cases, the Lemma holds. ∎

Corollary 2**.**

Let $AN_{n}$ be the $n$-dimensional alternating group network for $n\geq 6$. Then $t_{p}(AN_{n})=2n-5=\kappa_{1}(AN_{n})$.

Proof.   Note that $N=|V(AN_{n})|=\frac{n!}{2}\geq 4(n-1)-2$ for $n\geq 6$, Condition $(1)$ in Theorem 1 holds. Conditions $(2)$-$(4)$ in Theorem 1 hold by Lemmas 4, 5 and 22, respectively. So $AN_{n}$ satisfies all conditions in Theorem 1, and $t_{p}(AN_{n})=2n-5=\kappa_{1}(AN_{n})$ for $n\geq 6$. ∎

4.3 Application to BC networks

Note that $L_{n}=\{X_{n}:X_{n}\ {\rm is\ an}\ n-{\rm dimensional\ BC\ network}\}$. For a $BC$ network $X_{n}\in L_{n}$, the connectivity is $k=n\geq 5$, $l=0$, $N=|V|=2^{n}\geq 4n-2$ for $n\geq 5$ in Theorem 1. As a directive corollary of Theorem 1, we can get the result $\kappa_{1}(X_{n})=t_{p}(X_{n})=2n-2$ in which Zhu [47] determined $\kappa_{1}(X_{n})=2n-2$ for $n\geq 4$. Fan and Lin [19] obtained $t_{p}(X_{n})=2n-2$ for $n\geq 4$.

Lemma 23**.**

For any $X_{n}\in L_{n}$, if $U\subseteq V(X_{n})$ with $2\leq|U|\leq 4n-8$ for $n\geq 3$, then $|N_{X_{n}}(U)|\geq 2n-2$.

Proof.   We prove the lemma by using introduction on $n$. If $n=3$, $2\leq|U|\leq 4n-8=4$, it is not difficult to see that $|N_{X_{3}}(U)|\geq 4$. Assume that the lemma is true for $X_{m-1}$, where $m$ is an integer with $4\leq m\leq n-1$. We consider $X_{n}$ for $n\geq 4$ as follows.

Since $X_{n}$ is $n$-regular $n$-connected triangle-free and $C(X_{n})=2$, if $|U|=2$, then $|N_{X_{n}}(U)|\geq 2n-2$. Now consider $3\leq|U|\leq 4n-8$. Note that $X_{n}$ contains two copies of $X_{n-1}$, say $X_{n-1}^{1}$ and $X_{n-1}^{2}$, respectively. Let $U_{i}=U\cap V(X_{n-1}^{i})$ for $i\in\{1,2\}$. Without loss of generality, we may assume that $|U_{1}|\geq|U_{2}|$. It implies that $2\leq|U_{1}|$.

Case 1. $2\leq|U_{1}|\leq 4n-12$. By the inductive hypothesis in $X_{n-1}^{1}$, $|N_{X_{n-1}^{1}}(U_{1})|\geq 2n-4$. If $|U_{2}|=0$, then $U=U_{1}$. $|N_{X_{n}}(U)|\geq|N_{X_{n-1}^{1}}(U_{1})|+|N_{\overline{X_{n-1}^{1}}}(U_{1})|\geq(2n-4)+2\geq 2n-2$. If $|U_{2}|=1$, $|N_{X_{n-1}^{2}}(U_{2})|=\kappa(X_{n-1}^{2})=n-1$. Thus $|N_{X_{n}}(U)|\geq|N_{X_{n-1}^{1}}(U_{1})|+|N_{X_{n-1}^{2}}(U_{2})|\geq(2n-4)+(n-1)=3n-5\geq 2n-2$ for $n\geq 4$. Now consider $2\leq|U_{2}|\leq|U_{1}|\leq 4n-12$ for $n\geq 4$, so $|N_{X_{n-1}^{2}}(U_{2})|\geq 2n-4$. Thus, $|N_{X_{n}}(U)|\geq|N_{X_{n-1}^{1}}(U_{1})|+|N_{X_{n-1}^{2}}(U_{2})|\geq 2(2n-4)=4n-8\geq 2n-2$ for $n\geq 4$.

Case 2. $4n-11\leq|U_{1}|\leq 4n-8$.

If $U=U_{1}$, by definition, $|N_{\overline{X_{n-1}^{1}}}(U_{1})|=|U_{1}|\geq 4n-11$. Thus, $|N_{X_{n}}(U)|\geq|N_{\overline{X_{n-1}^{1}}}(U_{1})|\geq 4n-11\geq 2n-4$ for $n\geq 4$. Now assume that $U\neq U_{1}$. Since the connectivity of $X_{n-1}^{1}$ is $n-1$ and $|V(X_{n-1}^{1})|-(4n-8)\geq\kappa(X_{n-1}^{1})=n-1$ for $n\geq 4$, by Lemma 21, $|N_{X_{n-1}^{1}}(U_{1})|\geq n-1$. Note that $U\neq U_{1}$ and $|U-U_{1}|\leq 3$, so $1\leq|U_{2}|\leq 3$. If $|U_{2}|=1$, $|N_{X_{n-1}^{2}}(U_{2})|=\kappa(X_{n-1}^{2})=n-1$. Hence, $|N_{X_{n}}(U)|\geq|N_{X_{n-1}^{1}}(U_{1})|+|N_{X_{n-1}^{2}}(U_{2})|\geq 2n-2$ for $n\geq 4$. Now suppose that $2\leq|U_{2}|\leq 3$. Since $|V(X_{n-1}^{2})|-3\geq\kappa(B_{2})=n-1$ for $n\geq 4$, by Lemma 21, $|N_{X_{n-1}^{2}}(U_{2})|\geq\kappa(X_{n-1}^{2})=n-1$. So $|N_{X_{n}}(U)|\geq|N_{X_{n-1}^{1}}(U_{1})|+|N_{X_{n-1}^{2}}(U_{2})|\geq 2n-2$ for $n\geq 4$.

By the above cases, the proof is completed. ∎

By Lemmas 6, 7 and 23 and Theorem 1, we obtain the following Corollary 3.

Corollary 3**.**

For any $X_{n}\in L_{n}$, $t_{p}(X_{n})=2n-2=\kappa_{1}(X_{n})$ for $n\geq 5$.

It is not difficult to check that the hypercube $Q_{n}$, the crossed cube $CQ_{n}$, the M$\ddot{{\rm o}}$bius cubes $MQ_{n}$, the twisted cubes $TQ_{n}$ are all $n$-regular $n$-connected triangle-free BCs, then the following known result is derived directly.

Corollary 4**.**

([19])* Every pessimistic diagnosability of the hypercube $Q_{n}$, the crossed cube $CQ_{n}$, the M$\ddot{o}$bius cubes $MQ_{n}$ and the twisted cubes $TQ_{n}$ is $2n-2$ for $n\geq 6$..*

4.4 Application to the $k$-ary $n$-cube networks $Q_{n}^{k}$

Lemma 24**.**

Let $Q_{n}^{k}$ be a $k$-ary $n$-cube, where $k\geq 2$ and $n\geq 1$ are integers.

1. (1)

For $n\geq 3$, let $U$ be a subset of $V(Q_{n}^{2})$ with $2\leq|U|\leq 4n-8$. Then $|N_{Q_{n}^{2}}(U)|\geq 2n-2$. 2. (2)

For $n\geq 3$, let $U$ be a subset of $V(Q_{n}^{3})$ and $2\leq|U|\leq 8n-10$, then $|N_{Q_{n}^{3}}(U)|\geq 4n-3$. 3. (3)

For $n\geq 3$ and $k\geq 4$, let $U$ be a subset of $V(Q_{n}^{k})$ and $2\leq|U|\leq 8n-8$, then $|N_{Q_{n}^{k}}(U)|\geq 4n-2$.

Proof.   Since the proof for the three cases are similar, we take $(2)$ as an example, the details for $(1)$ and $(3)$ are omitted.

Let $Q[0],Q[1],Q[2]$ represent the three disjoint subcubes obtained from $Q_{n}^{3}$ by partition over one dimension. Let $U_{i}=U\cap V(Q[i])$ and $\overline{Q[i]}=Q_{n}^{3}-Q[i]$ for $i\in\{0,1,2\}$. Without loss of generality, we may assume that $|U_{0}|\geq|U_{1}|\geq|U_{2}|$.

The lemma is proved by the induction on $n$. When $n=3$, it is easy to check $|N_{Q_{3}^{3}}(U)|\geq 9$ for $2\leq|U|\leq 8n-10=14$. We assume that the lemma is true for $Q_{m-1}^{3}$, where $m$ is an integer with $4\leq m\leq n-1$. We consider $Q_{n}^{3}$ for $n\geq 4$ as follows.

Case 1. $|U_{0}|\leq 1$.

In this case, $|U_{i}|\leq 1$ for all $0\leq i\leq 2$. Clearly, $2\leq|U|\leq 3$ because of $i\leq 2$. The Lemma follows if $|U|=2$ by Lemma 19. Now assume that $|U|=3$. Since $Q_{n}^{3}$ is $2n$-regular and $Q[i]$ is isomorphic to $Q_{n-1}^{3}$, $|N_{Q_{n}^{3}}(U)|\geq 3\kappa(Q_{n-1}^{3})=3(2n-2)\geq 4n-3$ for $n\geq 3$.

Case 2. $2\leq|U_{0}|\leq 8n-18$.

By inductive hypothesis in $Q[0]$, $|N_{Q[0]}(U_{0})|\geq 4(n-1)-3=4n-7$. If $U=U_{0}$, then $|N_{Q_{n}^{3}}(U)|=|N_{Q[0]}(U_{0})|+|N_{\overline{Q[0]}}(U_{0})|\geq 4n-7+2|U_{0}|\geq 4n-7+4=4n-3$. Assume $U\neq U_{0}$ in the following. Note that $|U|\leq 8n-10$ and $|U_{0}|\geq|U_{1}|\geq|U_{2}|$, $|U_{1}|\leq 4n-5$.

If $|U_{1}|=1$, $|N_{Q[1]}(U_{1})|=\kappa(Q[1])=2n-2$. Note that $Q[0]$ and $Q[1]$ are vertex disjoint, $|N_{Q_{n}^{3}}(U)|\geq|N_{Q[0]}(U_{0})|+|N_{Q[1]}(U_{1})|\geq(4n-7)+(2n-2)=6n-9\geq 4n-3$ for $n\geq 4$. Now consider $2\leq|U_{1}|\leq 4n-5\leq 8n-18$ for $n\geq 4$, by inductive hypothesis in $Q[1]$, $|N_{Q[1]}(U_{1})|\geq 4(n-1)-3=4n-7$. Thus, $|N_{Q_{n}^{3}}(U)|\geq|N_{Q[0]}(U_{0})|+|N_{Q[1]}(U_{1})|\geq 2(4n-7)=8n-14\geq 4n-3$ for $n\geq 4$.

Case 3. $8n-17\leq|U_{0}|\leq 8n-10$.

If $U=U_{0}$, $|N_{Q_{n}^{3}}(U)|\geq|N_{\overline{Q[0]}}(U_{0})|=2|U_{0}|\geq 2(8n-17)\geq 4n-3$ for $n\geq 4$. In the following, we assume the case of $U\neq U_{0}$. Since the connectivity of $Q[0]$ is $2n-2$, note that $U\neq U_{0}$, so $2\leq|U_{0}|\leq 8n-11$. Since $|V(Q_{[0]})-U_{0}|=3^{n-1}-|U_{0}|\geq 3^{n-1}-(8n-11)\geq 2n-2=\kappa(Q[0])$ for $n\geq 4$, and by Lemma 21, $|N_{Q[0]}(U_{0})|\geq 2n-2$.

Note that $U\neq U_{0}$ and $|U-U_{0}|\leq 7$, so $1\leq|U_{1}|\leq 7$.

If $|U_{1}|=1$ and $|U_{2}|=0$, recall that the connectivity of $Q_{n}^{3}$ is $2n$ and $Q[i]$ is isomorphic to $Q_{n-1}^{k}$, $|N_{Q[1]}(U_{1})|=\kappa(Q[1])=2n-2$. Note that each vertex in $Q[0]$ (resp. $Q[1]$) has an extra neighbor in $Q[2]$. Hence, $|N_{Q_{n}^{3}}(U)|\geq|N_{Q[0]}(U_{0})|+|N_{Q[1]}(U_{1})|+|N_{Q[2]}(U_{0})|\geq 4n-4+(8n-17)=12n-21\geq 4n-3$ for $n\geq 4$. If $|U_{i}|=1$ for $i=1,2$, $|N_{Q[i]}(U_{i})|=\kappa(Q[i])=2n-2$. Hence, $|N_{Q_{n}^{3}}(U)|\geq|N_{Q[0]}(U_{0})|+|N_{Q[1]}(U_{1})|+|N_{Q[2]}(U_{2})|\geq 3(2n-2)=6n-6\geq 4n-3$ for $n\geq 4$. Now suppose that $2\leq|U_{1}|\leq 7$. Since $7<8n-17$ for $n\geq 4$, by inductive hypothesis in $Q[1]$, $|N_{Q[1]}(U_{1})|\geq 4(n-1)-3=4n-7$. Thus, $|N_{Q_{n}^{3}}(U)|\geq|N_{Q[0]}(U_{0})|+|N_{Q[1]}(U_{1})|\geq(2n-2)+(4n-7)=6n-9\geq 4n-3$ for $n\geq 4$.

The proof is complete. ∎

Remark 2**.**

Esfahanian [15] obtained $\kappa_{1}(Q_{n}^{2})=2n-2$ for $n\geq 3$ and Day [13] got $\kappa_{1}(Q_{n}^{3})=4n-3$, $\kappa_{1}(Q_{n}^{k})=4n-2$ for $k\geq 4$. Kavianpour and Kim [30] proved that $t_{p}(Q_{n}^{2})=2n-2$ for $n\geq 3$ and Wang et al. [41] derived $t_{p}(Q_{n}^{3})=4n-3$, $t_{p}(Q_{n}^{k})=4n-2$ for $k\geq 4$ and $n\geq 4$. These results can be gotten directly as corollary of Theorem 1 as following.

Since $k^{n}\geq 4\kappa(Q_{n}^{k})-2$ for $k\geq 3$ and $n\geq 3$ ($k=2$ and $n\geq 5$), Condition $(1)$ in Theorem 1 holds. By Lemmas 8, 9 and 24, Condition $(2)$-$(4)$ in Theorem 1 holds.

Corollary 5**.**

Let $Q_{n}^{k}$ be a $k$-ary $n$-cube, where $k\geq 2$ and $n\geq 1$ are integers. Then

1. (1)

$t_{p}(Q_{n}^{2})=2n-2=\kappa_{1}(Q_{n}^{2})$* for $n\geq 5$;* 2. (2)

$t_{p}(Q_{n}^{3})=4n-3=\kappa_{1}(Q_{n}^{3})$* for $n\geq 3$;* 3. (3)

$t_{p}(Q_{n}^{k})=4n-2=\kappa_{1}(Q_{n}^{k})$* for $n\geq 3$ and $k\geq 4$.*

4.5 Application to the split-star networks $S_{n}^{2}$

Lin et al. [32] proved $\kappa_{1}(S_{n}^{2})=4n-9$ for $n\geq 4$. However, $t_{p}(S_{n}^{2})$ has not been determined so far. We can deduce the result by Theorem 1 in which for $S_{n}^{2}$, $k=2n-3$, $l=1$.

Lemma 25**.**

Let $S_{n}^{2}$ be the $n$-dimensional split-star network for $n\geq 4$. If $U$ is a subset of $V(S_{n}^{2})$ and $2\leq|U|\leq 8n-22$, then $|N_{S_{n}^{2}}(U)|\geq 4n-9$.

Proof.   We prove the lemma by using the induction on $n$. Since $S_{4}^{2}$ is constructed by four disjoint triangles $S_{3}^{2}$, it is easy to verify that $|N_{S_{4}^{2}}(U)|\geq 7$ for $2\leq|U|\leq 10$. By the inductive hypothesis, we assume that the lemma is true for $S_{m}^{2}$, where $m$ is an integer with $5\leq m\leq n-1$. Now we consider $S_{n}^{2}$.

Recall that $S_{n}^{2}$ is constructed by $n$ disjoint $S_{n-1}^{2}$s, denoted by $S_{n}^{2:i}$ for $i\in[n]$. Let $U_{i}=U\cap V(S_{n}^{2:i})$ and $\overline{S_{n}^{2:i}}=S_{n}^{2}-S_{n}^{2:i}$ for $i\in[n]$. Without loss of generality, we may assume that $|U_{1}|\geq|U_{2}|\geq\ldots\geq|U_{n}|$. The following cases should be considered.

Case 1. $|U_{1}|\leq 1$.

In this case, $|U_{i}|\leq 1$ for all $i\in[n]$. Clearly, $2\leq|U|\leq n$ because of $U=\bigcup\limits_{i=1}^{n}U_{i}$. If $|U|=2$, by Lemma 19, $|N_{S_{n}^{2}}(U)|\geq 2(2n-3)-2-1=4n-9$, the lemma follows. Now assume that $3\leq|U|\leq n$. Since $S_{n}^{2}$ is $(2n-3)$-regular and $S_{n}^{2:i}$ is isomorphic to $S_{n-1}^{2}$, $|N_{S_{n}^{2}}(U)|\geq 3\kappa(S_{n}^{2:i})=3(2n-5)\geq 4n-9$ for $n\geq 5$.

Case 2. $2\leq|U_{1}|\leq 8n-30$.

By inductive hypothesis in $S_{n}^{2:1}$, $|N_{S_{n}^{1}}(U_{1})|\geq 4(n-1)-9=4n-13$. Since $|U|\leq 8n-22$ and $|U_{1}|\geq|U_{2}|\geq\ldots\geq|U_{n}|$, $|U_{2}|\leq 4n-11$. If $U=U_{1}$, by Lemma 10(2), $|N_{S_{n}^{2}}(U)|=|N_{S_{n}^{2:1}}(U_{1})|+|N_{\overline{S_{n}^{2:1}}}(U_{1})|\geq 4n-13+2|U_{1}|\geq 4n-9$. Assume $U\neq U_{1}$ in the following. If $|U_{2}|=1$, $|N_{S_{n}^{2:1}}(U_{1})|=\kappa(S_{n}^{2:1})=2n-5$. Note that $S_{n}^{2:1}$ and $S_{n}^{2:2}$ are vertex disjoint, $|N_{S_{n}^{2}}(U)|\geq|N_{S_{n}^{2:1}}(U_{1})|+|N_{S_{n}^{2:2}}(U_{2})|\geq 4n-13+2n-5=6n-18\geq 4n-9$ for $n\geq 5$. Now consider $2\leq|U_{2}|\leq 4n-11$. Note that $4n-11\leq 8n-30$ for $n\geq 5$, by inductive hypothesis in $S_{n}^{2:2}$, $|N_{S_{n}^{2:2}}(U_{2})|\geq 4(n-1)-9=4n-13$. Thus, $|N_{S_{n}^{2}}(U)|\geq|N_{S_{n}^{2:1}}(U_{1})|+|N_{S_{n}^{2:2}}(U_{2})|\geq 8n-26\geq 4n-9$ for $n\geq 5$.

Case 3. $8n-29\leq|U_{1}|\leq 8n-22$.

By Lemma 10(2), $|N_{\overline{S_{n}^{2:1}}}(U_{1})|=2|U_{1}|$. If $U=U_{1}$, $|N_{S_{n}^{2}}(U)|\geq|N_{\overline{S_{n}^{2:1}}}(U_{1})|=2|U_{1}|\geq 16n-58\geq 4n-9$ for $n\geq 5$. In the following, we assume the case of $U\neq U_{1}$. Since the connectivity of $S_{n}^{2:1}$ is $2n-5$, and $(n-1)!-|U_{1}|\geq 2n-5=\kappa(S_{n}^{2:1})$ for $n\geq 5$, by Lemma 21, $|N_{S_{n}^{2:1}}(U_{1})|\geq 2n-5$. Note that $U\neq U_{1}$ and $|U-U_{1}|\leq 7$, so $1\leq|U_{2}|\leq 7$.

If $|U_{2}|=1$, recall that $S_{n}^{2}$ is $(2n-3)$-regular and $S_{n}^{2:i}$ is isomorphic to $S_{n-1}^{2}$, $|N_{S_{n}^{2:2}}(U_{2})|=\kappa(S_{n}^{2:2})=2n-5$. Hence, $|N_{S_{n}^{2}}(U)|\geq|N_{\overline{S_{n}^{2:1}}}(U_{1})|-|U-U_{1}|\geq 16n-65\geq 4n-9$ for $n\geq 5$. Now suppose that $2\leq|U_{2}|\leq 7$. Since $7\leq 8n-30$ for $n\geq 5$, by inductive hypothesis in $S_{n}^{2:1}$, $|N_{S_{n}^{2:1}}(U_{1})|\geq 4(n-1)-9=4n-13$. Thus, $|N_{S_{n}^{2}}(U)|\geq|N_{S_{n}^{2:1}}(U_{1})|+|N_{S_{n}^{2:2}}(U_{2})|\geq(2n-5)+(4n-13)=6n-18\geq 4n-9$ for $n\geq 5$.

By the above cases, the lemma holds. ∎

Corollary 6**.**

Let $S_{n}^{2}$ be the $n$-dimensional split-star network for $n\geq 4$. Then $t_{p}(S_{n}^{2})=4n-9=\kappa_{1}(S_{n}^{2})$.

Proof.   To prove the theorem, we only need to verify that $S_{n}^{2}$ satisfies conditions in Theorem 1. Note that $k=2n-3\geq 5$ for $n\geq 4$, $l=1$, $N=|V(S_{n}^{2})|=n!\geq 4(2n-3)-2$ for $n\geq 4$, Condition $(1)$ in Theorem 1 holds. By Lemmas 10 and 25, Conditions $(2)$-$(3)$ in Theorem 1 holds. Condition $(4)$ holds by Lemma 11. $S_{n}^{2}$ satisfies all conditions in Theorem 1, and thus $t_{p}(S_{n}^{2})=4n-9=\kappa_{1}(S_{n}^{2})$. ∎

4.6 Application to the Cayley graphs generated by transposition trees $\Gamma_{n}$

Let $\Gamma_{n}$ be Cayley graphs generated by transposition trees. Yang et al. [42] determined $\kappa_{1}(\Gamma_{n})=2n-4$ for $n\geq 3$. However, $t_{p}(\Gamma_{n})$ has not been known so far. By Theorem 1, we immediately the following result which contains the above result. Note that for $\Gamma_{n}$, $k=n-1$, $l=0$ in Theorem 1.

Lemma 26**.**

Let $\Gamma_{n}$ be Cayley graphs generated by transposition trees for $n\geq 4$. If $U$ is a subset of $V(\Gamma_{n})$ and $2\leq|U|\leq 4n-12$, then $|N_{\Gamma_{n}}(U)|\geq 2n-4$.

Proof.   The lemma is proved by induction on $n$. When $n=4$, it is easy to check $|N_{\Gamma_{n}}(U)|\geq 4$ for $2\leq|U|\leq 4n-12=4$. We assume that the lemma is true for $\Gamma_{m}$, where $m$ is an integer with $4\leq m\leq n-1$. We consider $\Gamma_{n}$ for $n\geq 5$ as follows.

Recall that $\Gamma_{n}$ can be decomposed into $n$ copies of $\Gamma_{n-1}^{\prime}s$, namely $\Gamma_{n-1}^{1},\Gamma_{n-1}^{2},\ldots,\Gamma_{n-1}^{n}$. Let $U_{i}=U\cap V(\Gamma_{n-1}^{i})$ and $\overline{\Gamma_{n-1}^{i}}=\Gamma_{n}-\Gamma_{n-1}^{i}$ for $i\in[n]$. Without loss of generality, we may assume that $|U_{1}|\geq|U_{2}|\geq|U_{3}|\geq\ldots\geq|U_{n}|$.

Case 1. $|U_{1}|\leq 1$.

In this case, $|U_{i}|\leq 1$ for all $1\leq i\leq n$. Since $|U|\geq 2$, it implies $|U_{1}|=|U_{2}|=1$. Since $\Gamma_{n}$ is $(n-1)$-regular and $\Gamma_{n-1}^{i}$ is isomorphic to $\Gamma_{n-1}$, $|N_{\Gamma_{n}}(U)|\geq 2\kappa(\Gamma_{n-1}^{i})=2(n-2)=2n-4$ for $n\geq 5$.

Case 2. $2\leq|U_{1}|\leq 4n-16$.

By inductive hypothesis in $\Gamma_{n-1}^{1}$, $|N_{\Gamma_{n-1}^{1}}(U_{1})|\geq 2(n-1)-4=2n-6$. Note that $|U_{i}|\leq|U_{1}|\leq 4n-16$ for $i\in\{2,3,\ldots,n\}$. If $|U_{2}|=1$, $|N_{\Gamma_{n-1}^{2}}(U_{2})|\geq\kappa(\Gamma_{n-1}^{2})=n-2$, so $|N_{\Gamma_{n}}(U)|\geq|N_{\Gamma_{n-1}^{1}}(U_{1})|+|N_{\Gamma_{n-1}^{2}}(U_{2})|\geq(2n-6)+(n-2)=3n-8\geq 2n-4$ for $n\geq 5$. If $2\leq|U_{2}|\leq 4n-16$, by inductive hypothesis in $\Gamma_{n-1}^{2}$, $|N_{\Gamma_{n-1}^{2}}(U_{2})|\geq 2(n-1)-4=2n-6$. Thus, $|N_{\Gamma_{n}}(U)|\geq|N_{\Gamma_{n-1}^{1}}(U_{1})|+|N_{\Gamma_{n-1}^{2}}(U_{2})|\geq 2(2n-6)=4n-12\geq 2n-4$ for $n\geq 5$. Now consider $|U_{2}|=0$, then $|U_{i}|=0$ for $i\in\{3,4,\ldots,n\}$, it implies that $U=U_{1}$. So $|N_{\Gamma_{n}}(U)|\geq|N_{\Gamma_{n-1}^{1}}(U_{1})|+|N_{\overline{\Gamma_{n-1}^{1}}}(U_{1})|\geq 2n-6+|U_{1}|\geq 2n-6+2=2n-4$ for $n\geq 5$.

Case 3. $4n-15\leq|U_{1}|\leq 4n-12$.

If $U=U_{1}$, by Lemma 12, $|N_{\overline{\Gamma_{n-1}^{1}}}(U_{1})|=|U_{1}|\geq 4n-15$. Since $(n-1)!-(4n-12)\geq n-2$ for $n\geq 5$, by Lemma 21, $|N_{\Gamma_{n-1}^{1}}(U_{1})|\geq\kappa(\Gamma_{n-1}^{1})=n-2$. Thus, $|N_{\Gamma_{n}}(U)|=|N_{\overline{\Gamma_{n-1}^{1}}}(U_{1})|+|N_{\Gamma_{n-1}^{1}}(U_{1})|\geq 4n-15+(n-2)=5n-17\geq 2n-4$ for $n\geq 5$. In the following, we assume that $U\neq U_{1}$. It implies that $|U-U_{1}|\leq 3$, so $1\leq|U_{2}|\leq|U|-|U_{1}|\leq 3$.

If $|U_{2}|=1$, $|N_{\Gamma_{n-1}^{2}}(U_{2})|=\kappa(\Gamma_{n-1}^{2})=n-2$. Recall that $|N_{\Gamma_{n-1}^{1}}(U_{1})|\geq n-2$. Hence, $|N_{\Gamma_{n}}(U)|\geq|N_{\Gamma_{n-1}^{0}}(U_{0})|+|N_{\Gamma_{n-1}^{1}}(U_{1})|\geq 2n-4$ for $n\geq 5$. Now suppose that $2\leq|U_{2}|\leq 3$. Since $|U_{2}|\leq 3\leq 4n-16$ for $n\geq 5$, by inductive hypothesis in $\Gamma_{n-1}^{2}$, $|N_{\Gamma_{n-1}^{2}}(U_{2})|\geq 2(n-1)-4=2n-6$. Thus, $|N_{\Gamma_{n}}(U)|\geq|N_{\Gamma_{n-1}^{1}}(U_{1})|+|N_{\Gamma_{n-1}^{2}}(U_{2})|\geq(n-2)+(2n-6)=3n-8\geq 2n-4$ for $n\geq 5$.

By the above cases, the proof is completed. ∎

Corollary 7**.**

Let $\Gamma_{n}$ be Cayley graphs generated by transposition trees for $n\geq 6$. Then $t_{p}(\Gamma_{n})=2n-4=\kappa_{1}(\Gamma_{n})$ for $n\geq 6$.

Proof.   Note that $k=n-1\geq 5$ and $N=|V(\Gamma_{n})|=n!\geq 4(n-1)-2$ for $n\geq 6$, Condition $(1)$ in Theorem 1 holds. By Lemma 12 and 26, Condition $(2)$-$(3)$ in Theorem 1 holds. Condition $(4)$ holds by Lemma 13. Thus, $\Gamma_{n}$ satisfies all conditions in Theorem 1, $t_{p}(\Gamma_{n})=2n-4=\kappa_{1}(\Gamma_{n})$ for $n\geq 6$. ∎

Since the star graph and the bubble-sort graph are Cayley graph generated by transposition trees, The following corollary is gotten directly from Corollary 7.

Corollary 8**.**

Let $S_{n}$ and $B_{n}$ are the star graph and the bubble sort graph, then $t_{p}(S_{n})=2n-4=\kappa_{1}(S_{n})$ for $n\geq 6$, and $t_{p}(B_{n})=2n-4=\kappa_{1}(B_{n})$ for $n\geq 6$.

4.7 Application to the Cayley graphs generated by $2$-trees $\Gamma_{n}(\Delta)$

Lemma 27**.**

Let $\Gamma_{n}(\Delta)$ be a Cayley graph generated by the $2$-tree $\Delta$. For $n\geq 4$, let $U$ be a subset of $V(\Gamma_{n}(\Delta))$ and $2\leq|U|\leq 8n-26$. Then, $|N_{\Gamma_{n}(\Delta)}(U)|\geq 4n-11$.

Proof.   The lemma is proved by the induction on $n$. Since $\Gamma_{4}(\Delta)$ is constructed by $4$ disjoint triangles, it is easy to verify that $|N_{\Gamma_{4}(\Delta)}(U)|\geq 5$ for $2\leq|U|\leq 7$. By the inductive hypothesis, we assume that the lemma is true for $\Gamma_{m}(\Delta)$, where $m$ is an integer with $5\leq m\leq n-1$.

Note that $\Gamma_{n}(\Delta)$ is constructed by $n$ disjoint $\Gamma_{n-1}(\Delta)$, denoted by $\Gamma_{n}^{i}(\Delta)$ for $i\in[n]$. Let $U_{i}=U\cap V(\Gamma_{n-1}^{i}(\Delta))$ and $\overline{\Gamma_{n-1}^{i}(\Delta)}=\Gamma_{n}(\Delta)-\Gamma_{n-1}^{i}(\Delta)$ for $i\in[n]$. Without loss of generality, we may assume that $|U_{1}|\geq|U_{2}|\geq\ldots\geq|U_{n}|$. The following three cases should be considered.

Case 1. $|U_{1}|\leq 1$.

In this case, $|U_{i}|\leq 1$ for all $i\in[n]$. Clearly, $2\leq|U|\leq n$ because of $i\leq n$. The Lemma follows if $|U|=2$ by Lemma 19. Now assume that $3\leq|U|\leq n$. Since $\Gamma_{n}(\Delta)$ is $(2n-4)$-regular and $\Gamma_{n-1}^{i}(\Delta)$ is isomorphic to $\Gamma_{n-1}(\Delta)$, $|N_{\Gamma_{n}(\Delta)}(U)|\geq 3\kappa(\Gamma_{n-1}^{i}(\Delta))=3(2n-6)\geq 4n-11$ for $n\geq 5$.

Case 2. $2\leq|U_{1}|\leq 8n-34$.

By inductive hypothesis in $\Gamma_{n-1}^{1}(\Delta)$, $|N_{\Gamma_{n-1}^{1}(\Delta)}(U_{1})|\geq 4(n-1)-11=4n-15$. If $U=U_{1}$, $|N_{\Gamma_{n}(\Delta)}(U)|=|N_{\Gamma_{n-1}^{1}(\Delta)}(U_{1})|+|N_{\overline{\Gamma_{n-1}^{1}(\Delta)}}(U_{1})|\geq 4n-15+2|U_{1}|\geq 4n-11$. Assume $U\neq U_{1}$ in the following. If $|U_{2}|=1$, $|N_{\Gamma_{n-1}^{2}(\Delta)}(U_{2})|=\kappa(\Gamma_{n-1}^{2}(\Delta))=2n-6$. Note that $\Gamma_{n-1}^{1}(\Delta)$ and $\Gamma_{n-1}^{2}(\Delta)$ are vertex disjoint, $|N_{\Gamma_{n}(\Delta)}(U)|\geq|N_{\Gamma_{n-1}^{1}(\Delta)}(U_{1})|+|N_{\Gamma_{n-1}^{2}(\Delta)}(U_{2})|\geq 4n-15+(2n-6)\geq 6n-21$ for $n\geq 5$. Now consider $2\leq|U_{2}|\leq|U_{1}|\leq 8n-34$, by inductive hypothesis in $\Gamma_{n-1}^{2}(\Delta)$, $|N_{\Gamma_{n-1}^{2}(\Delta)}(U_{2})|\geq 4(n-1)-11=4n-15$. Thus, $|N_{\Gamma_{n}(\Delta)}(U)|\geq|N_{\Gamma_{n-1}^{1}(\Delta)}(U_{1})|+|N_{\Gamma_{n-1}^{2}(\Delta)}(U_{2})|\geq 8n-30\geq 4n-11$ for $n\geq 5$.

Case 3. $8n-33\leq|U_{1}|\leq 8n-26$.

By Lemma 14, $|N_{\overline{\Gamma_{n-1}^{1}(\Delta)}}(U_{1})|=2|U_{1}|$. It is clear that the lemma holds if $U=U_{1}$. In the following, we assume the case of $U\neq U_{1}$. Since the connectivity of $\Gamma_{n-1}^{1}(\Delta)$ is $2n-6$, and by Lemma 21, $|N_{\Gamma_{n-1}^{1}(\Delta)}(U_{1})|\geq 2n-6$. Note that $U\neq U_{1}$ and $|U-U_{1}|\leq 7$, so $1\leq|U_{2}|\leq 7$.

If $|U_{2}|=1$, $|N_{\Gamma_{n}(\Delta)}(U)|\geq|N_{\Gamma_{n-1}^{1}(\Delta)}(U_{1})|+|N_{\overline{\Gamma_{n-1}^{1}(\Delta)}}(U_{1})|-|U-U_{1}|\geq(2n-6)+2|U_{1}|-7\geq 18n-79\geq 4n-11$ for $n\geq 5$. Now suppose that $2\leq|U_{2}|\leq 7$. Since $7\leq 8n-32$ for $n\geq 5$, by inductive hypothesis in $\Gamma_{n-1}^{2}(\Delta)$, $|N_{\Gamma_{n-1}^{2}(\Delta)}(U_{2})|\geq 4(n-1)-11=4n-15$. Thus, $|N_{\Gamma_{n}(\Delta)}(U)|\geq|N_{\Gamma_{n-1}^{1}(\Delta)}(U_{1})|+|N_{\Gamma_{n-1}^{2}(\Delta)}(U_{2})|\geq(2n-6)+(4n-15)=6n-21\geq 2n-5$ for $n\geq 5$.

By the above cases, the lemma holds. ∎

Corollary 9**.**

Let $G=\Gamma_{n}(\Delta)$ be a Cayley graph generated by the $2$-tree $\Delta$ for $n\geq 5$. Then $\kappa_{1}(G)=4n-11=t_{p}(G)$.

Proof.   Note that $k=2n-4\geq 5$ and $\frac{n!}{2}\geq 4(2n-4)-2$ for $n\geq 5$, Condition $(1)$ in Theorem 1 holds. By Lemma 14 and 27, Condition $(2)$ and $(3)$ in Theorem 1 holds. Condition $(4)$ holds by $|F|\leq 2k-3-l=2(2n-4)-3-1=4n-12<4n-11$ and Lemma 16. Thus, $\Gamma_{n}(\Delta)$ satisfies all conditions in Theorem 1, and so $t_{p}(\Gamma_{n}(\Delta))=4n-11=\kappa_{1}(\Gamma_{n}(\Delta))$ for $n\geq 5$. ∎

4.8 Application to the burnt pancake networks $BP_{n}$

Lemma 28**.**

Let $BP_{n}$ be the $n$-dimensional burnt pancake network. For $n\geq 3$, let $U$ be a subset of $V(BP_{n})$ and $2\leq|U|\leq 4n-8$, then $|N_{BP_{n}}(U)|\geq 2n-2$.

Proof.   If $|U|=2$, by Lemma 17 and Lemma 19, for any two distinct vertices $u$ and $v$, so $|N_{BP_{n}}(U)|\geq 2n-2$.

Recall that $BP_{n}$ can be decomposed into $2n$ copies of $BP_{n-1}$’s, namely $BP_{n-1}^{i}$, for $i\in\langle n\rangle$. Let $U_{i}=U\cap V(BP_{n-1}^{i})$ and $\overline{BP_{n-1}^{i}}=BP_{n}-BP_{n-1}^{i}$ for $i\in\langle n\rangle$. Without loss of generality, we may assume that $|U_{1}|\geq|U_{2}|\geq|U_{3}|\geq\ldots\geq|U_{n}|\geq|U_{\bar{n}}|\geq|U_{\overline{n-1}}|\geq|U_{\bar{1}}|$.

The lemma is proved by using the induction on $n$. If $n=3$, it is easy to check $|N_{BP_{n}}(U)|\geq 4$ for $2\leq|U|\leq 4n-8=4$. We assume that the lemma is true for $BP_{m}$, where $m$ is an integer with $4\leq m\leq n-1$. We consider $BP_{n}$ for $n\geq 4$ as follows.

Case 1. $|U_{1}|\leq 1$.

In this case, $|U_{i}|\leq 1$ for all $1\leq i\leq n$. Since $|U|\geq 2$, it implies that $|U_{1}|=|U_{2}|=1$. Since $BP_{n}$ is $n$-regular and $BP_{n-1}^{i}$ is isomorphic to $BP_{n-1}$, $|N_{BP_{n}}(U)|\geq 2\kappa(BP_{n-1}^{i})=2(n-1)=2n-2$ for $n\geq 4$.

Case 2. $2\leq|U_{1}|\leq 4n-12$.

By inductive hypothesis in $BP_{n-1}^{1}$, $|N_{BP_{n-1}^{1}}(U_{1})|\geq 2(n-1)-2=2n-4$. Note that $|U_{i}|\leq|U_{1}|\leq 4n-12$ for $i\in[n]\setminus\{1\}$. If $U=U_{1}$, $|N_{BP_{n}}(U)|=|N_{BP_{n-1}^{1}}(U_{1})|+|N_{\overline{BP_{n-1}^{1}}}(U_{1})|\geq 4n-12+|U_{1}|\geq 4n-11$. Assume $U\neq U_{1}$ in the following. If $|U_{2}|=1$, $|N_{BP_{n-1}^{2}}(U_{2})|\geq\kappa(BP_{n-1}^{2})=n-1$, so $|N_{BP_{n}}(U)|\geq|N_{BP_{n-1}^{1}}(U_{1})|+|N_{BP_{n-1}^{2}}(U_{2})|\geq(2n-4)+(n-1)=3n-5\geq 2n-2$ for $n\geq 4$. If $2\leq|U_{2}|\leq 4n-12$, by inductive hypothesis in $BP_{n-1}^{2}$, $|N_{BP_{n-1}^{2}}(U_{2})|\geq 2(n-1)-2=2n-4$. Thus, $|N_{BP_{n}}(U)|\geq|N_{BP_{n-1}^{1}}(U_{1})|+|N_{BP_{n-1}^{2}}(U_{2})|\geq 2(2n-4)=4n-8\geq 2n-2$ for $n\geq 4$.

Case 3. $4n-11\leq|U_{1}|\leq 4n-8$.

Since $(n-1)!-(4n-8)\geq n-1$ for $n\geq 5$, by Lemma 21, $|N_{BP_{n-1}^{1}}(U_{1})|\geq\kappa(BP_{n-1}^{1})=n-1$. If $U=U_{1}$, by Lemma 17, $|N_{\overline{BP_{n-1}^{1}}}(U_{1})|=|U_{1}|\geq 4n-11$. Thus, $|N_{BP_{n}}(U)|=|N_{\overline{BP_{n-1}^{1}}}(U_{1})|+|N_{BP_{n-1}^{1}}(U_{1})|\geq 4n-11+(n-1)=5n-2\geq 2n-2$ for $n\geq 4$. In the following, we assume that $U\neq U_{1}$. It implies that $|U-U_{1}|\leq 3$, so $1\leq|U_{2}|\leq|U|-|U_{1}|\leq 3$.

If $|U_{2}|=1$, $|N_{BP_{n-1}^{2}}(U_{2})|=\kappa(BP_{n-1}^{2})=n-1$. Recall that $|N_{BP_{n-1}^{1}}(U_{1})|\geq n-1$. Hence, $|N_{BP_{n}}(U)|\geq|N_{BP_{n-1}^{1}}(U_{1})|+|N_{BP_{n-1}^{2}}(U_{2})|\geq 2n-2$ for $n\geq 4$. Now suppose that $2\leq|U_{2}|\leq 3$. Since $|U_{2}|\leq 3\leq 4n-12$ for $n\geq 4$, by inductive hypothesis in $BP_{n-1}^{2}$, $|N_{BP_{n-1}^{2}}(U_{2})|\geq 2(n-1)-2=2n-4$. Thus, $|N_{BP_{n}}(U)|\geq|N_{BP_{n-1}^{1}}(U_{1})|+|N_{BP_{n-1}^{2}}(U_{2})|\geq(n-1)+(2n-4)=3n-5\geq 2n-2$ for $n\geq 4$.

By the above cases, the proof is completed. ∎

Remark 3**.**

The extra connectivity of $BP_{n}$ was obtained by Song et al. [35], $\kappa_{1}(BP_{n})=2n-2$ for $n\geq 4$. But $t_{p}(BP_{n})$ is not known so far. By Theorem 1, we immediately the following result which contains the above result.

Corollary 10**.**

Let $BP_{n}$ be the $n$-dimensional burnt pancake network for $n\geq 5$. Then $t_{p}(BP_{n})=2n-2=\kappa_{1}(BP_{n})$.

Proof.   Note that $k=n\geq 5$ and $N=|V(BP_{n})|=n!\geq 4n-2$ for $n\geq 5$, Condition $(1)$ in Theorem 1 holds. By Lemmas 17 and 28, Conditions $(2)$ and $(3)$ in Theorem 1 hold. Condition $(4)$ holds by Lemma 18. $BP_{n}$ satisfies all conditions in Theorem 1, and so $t_{p}(BP_{n})=2n-2=\kappa_{1}(BP_{n})$ for $n\geq 5$. ∎

5 Concluding remarks

This paper establishes the close relationship between these two parameter: the extra connectivity and pessimistic diagnosability under the PMC model, by proving $t_{p}(G)=\kappa_{1}(G)$ for some regular graphs $G$ with some conditions. As applications, the pessimistic diagnosability for each of split-star networks $S_{n}^{2}$, Cayley graphs generated by transposition trees $\Gamma_{n}$, Cayley graph generated by the $2$-tree $\Gamma_{n}(\Delta)$ and the burnt pancake networks $BP_{n}$ is gotten. As corollaries, the known results about the extra connectivity and the pessimistic diagnosability of many famous networks including the alternating group graphs [33], [37], the alternating group networks [45] , BC networks [47], [19] and the $k$-ary $n$-cube networks [15], [13], [30], [41] are obtained directly.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.11371052, No.11271012 and No.11171020).