On $g$-good-neighbor conditional diagnosability of $(n, k)$-star networks

TL;DR

This paper extends the understanding of $g$-good-neighbor conditional diagnosability for $(n,k)$-star networks under the PMC and MM* models, covering all remaining cases and generalizing previous results.

Contribution

It determines the $g$-good-neighbor conditional diagnosability for all remaining cases of $(n,k)$-star networks, broadening prior partial results under two fault diagnosis models.

Findings

Complete characterization of $t_g(S_{n,k})$ for all $1 \leq g \leq n-1$ and $1 \leq k \leq n-1$.

Generalization of diagnosability results to the star graph case.

Enhanced understanding of fault diagnosis capabilities in complex network topologies.

Abstract

The $g$-good-neighbor conditional diagnosability is a new measure for fault diagnosis of systems. Xu et al. [Theor. Comput. Sci. 659 (2017) 53--63] determined the $g$-good-neighbor conditional diagnosability of $(n, k)$-star networks $S_{n, k}$ (i.e., $t_g(S_{n, k})$) with $1\leq k\leq n-1$ for $1\leq g\leq n-k$ under the PMC model and the MM$^*$ model. In this paper, we determine $t_g(S_{n, k})$ for all the remaining cases with $1\leq k\leq n-1$ for $1\leq g\leq n-1$ under the two models, from which we can obtain the $g$-good-neighbor conditional diagnosability of the star graph obtained by Li et al. [to appear in Theor. Comput. Sci.] for $1\leq g\leq n-2$.

Tables

Table 1. Table 1:   The g 𝑔 g -good-neighbor conditional diagnosability of S n , k subscript 𝑆 𝑛 𝑘 S_{n,k} under the PMC model and the MM ∗ model

	$k=1$	$k=2$	$3\le k\le n-2$	$k=n-1$
$g=1$	$\lceil \frac{n}{2}\rceil -1\left(n\ge 4\right)$ [25]	$n$ (PMC) [25]	$n+k-2$ [25]	$2n-3$ [14, 25]
	$?$ ($n=3$)	$?$ ($n\ge 3$, MM∗)
$2\le g\le \lfloor \frac{n}{2}\rfloor -1$	$\lceil \frac{n}{2}\rceil -1$ [25]	$n+g\left(k-1\right)-1$ [25]
$\lfloor \frac{n}{2}\rfloor \le g\le n-k$	$?$			$\left(n-g\right)\left(g+1\right)!-1$ [14]
$n-k\le g\le n-2$	Nonexistence	$?$
$g=n-1$	$0$

Table 2. Table 2:   The g 𝑔 g -good-neighbor conditional diagnosability of S n , k subscript 𝑆 𝑛 𝑘 S_{n,k} under the PMC model and the MM ∗ model

	$k=1$	$k=2$	$3\le k\le n-2$	$k=n-1$
	$\lceil \frac{n}{2}\rceil -1\left(n\ge 4\right)$ [25]	$n$ (PMC) [25]
$g=1$	$0$ ($n=3$, MM∗) (Theorem 3.6)	$n-1$ ($n\ge 4$, MM∗) (Theorem 3.7)	$n+k-2$ [25]	$2n-3$ [14, 25]
	$1$ ($n=3$, PMC) (Theorem 3.6)	$1$ ($n=3$, MM∗) (Theorem 3.7)
$2\le g\le \lfloor \frac{n}{2}\rfloor -1$	$\lceil \frac{n}{2}\rceil -1$ [25]	$n+g\left(k-1\right)-1$ [25]
$\lfloor \frac{n}{2}\rfloor \le g\le n-k$	$n-g-1$ (Theorem 3.6)			$\left(n-g\right)\left(g+1\right)!-1$ [14]
$n-k\le g\le n-2$	Nonexistence	$\frac{\left(g+1\right)!\left(n-g\right)}{\left(n-k\right)!}-1\left(n\ge 4\right)$ (Theorem 3.5)
$g=n-1$	$0$

Equations

$$t_{g}(S_{n,k})=\left\{\begin{array}[]{lll}\left\lceil\dfrac{n}{2}\right\rceil-1&\hbox{if $1\leq g\leq\left\lfloor\dfrac{n}{2}\right\rfloor-1,\ k=1,\ n\geq 4$};\\ \\ n+g(k-1)-1&\hbox{if $1\leq g\leq n-k,\ 2\leq k\leq n-1$},\end{array}\right.$$

$$t_{g}(S_{n,k})=\left\{\begin{array}[]{lll}\left\lceil\dfrac{n}{2}\right\rceil-1&\hbox{if $1\leq g\leq\left\lfloor\dfrac{n}{2}\right\rfloor-1,\ k=1,\ n\geq 4$};\\ \\ n+k-2&\hbox{if $g=1,\ 3\leq k\leq n-1,\ n\geq 4$};\\ \\ n+g(k-1)-1&\hbox{if $2\leq g\leq n-k,\ 2\leq k\leq n-1$},\end{array}\right.$$

$$\sigma(u,v)=\left\{\begin{array}[]{lll}0&\hbox{if $\{u,v\}\cap F=\emptyset$};\\ \\ 1&\hbox{if $u\notin F$, $v\in F$};\\ \\ 0~{}{\rm or}~{}1&\hbox{if $u\in F$,}\end{array}\right.$$

$$\sigma\big{(}(u,v)_{w}\big{)}=\left\{\begin{array}[]{lll}0&\hbox{if $\{u,v,w\}\cap F=\emptyset$};\\ \\ 1&\hbox{if $w\notin F$, $\{u,v\}\cap F\neq\emptyset$};\\ \\ 0~{}{\rm or}~{}1&\hbox{if $w\in F$,}\end{array}\right.$$

$$A=\{p_1{p}_2\dots p_{k-(n-g-1)}12\dots (n-g-1)\in V(S_{n,k})\mid p_i\in I_n-I_{n-g-1},i\in I_{k-(n-g-1)}\},$$

$$F_1=\{i{p}_2\dots p_{k-(n-g-1)}1\dots (i-1)p_1(i+1)\dots (n-g-1)\in V(S_{n,k})|p_j\in I_n-I_{n-g-1},j\in I_{n-g-1}\}.$$

$$N(u)\cap A$$

$$N(u)\cap F_1=\{i{p}_2\dots p_{k-(n-g-1)}1\dots (i-1)p_1(i+1)\dots (n-g-1)\in V(S_{n,k})\mid i\in I_{n-g-1}\}.$$

$$j{p}_2\dots p_{k-(n-g-1)}1\dots (j-1)i(j+1)\dots (i-1)p_1(i+1)\dots (n-g-1),j\in I_{n-g-1},i\ne j;$$

$$p_j{p}_2\dots p_{j-1}i{p}_{j+1}\dots p_{k-(n-g-1)}1\dots (i-1)p_1(i+1)\dots (n-g-1),j\in I_{k-(n-g-1)}^{\prime };$$

$$q{p}_2\dots p_{k-(n-g-1)}1\dots (i-1)p_1(i+1)\dots (n-g-1),q\in I_n-I_{n-g-1}-\{p_1,p_2,\dots ,p_{k-(n-g-1)}\}.$$

$$\begin{array}{rl}|V(S_{n,1})|-|F_1\cup F_2|& =|V(S_{n,1})|-|F_1|-|F_2|+|F_1\cap F_2|\\ & \ge n-2(n-g-1)+|F_1\cap F_2|\\ & =2(g+1)-n+|F_1\cap F_2|\\ & \ge 2(\lfloor \frac{n}{2}\rfloor +1)-n+|F_1\cap F_2|\\ & \ge 1+|F_1\cap F_2|.\end{array}$$

$$t_{1}(S_{n,2})=\left\{\begin{array}[]{lll}n-1&\hbox{if $n\geq 4$};\\ \\ 1&\hbox{if $n=3$}.\end{array}\right.$$

$$\sum _{w\in W}|N(w)\cap (F_1\cap F_2)|$$

$$|F_1|+|F_2|$$

$$\max \{|F_1|,|F_2|\}$$

$$A=N(\{12,32,42\}),$$

$$F_1=A\cup \{12\}$$

$$F_2=A\cup \{32\}.$$

Full text

Key words: PMC model; MM∗ model; $(n,k)$-star networks; Fault diagnosability.

1 Introduction

With the size of multiprocessor systems increasing, processor failure is inevitable. Thus, to evaluate the reliability of multiprocessor systems, fault diagnosability has become an important metric. Many models have been proposed for determining a multiprocessor system’s diagnosability. The PMC model was proposed by Preparata, Metze, and Chien [21] for fault diagnosis in multiprocessor systems. In the PMC model, all processors in the system under diagnosis can test one another. The MM model, proposed by Maeng and Malek [19], assumes that a vertex in the system sends the same task to two of its neighbors and then compares their responses. Sengupta and Dahbura [22] further suggested a modification of the MM model, called the MM∗ model, in which each processor has to test two processors if the processor is adjacent to the latter two processors. Many researchers have applied the PMC model and the MM∗ model to identify faults in various topologies (see [4, 5, 6, 7, 8, 11, 12, 14, 15, 18, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]).

The classical diagnosability for multiprocessor systems assumes that all the neighbors of any processor may fail simultaneously. However, the probability that this event occurs is very small in large-scale multiprocessor systems. In 2005, Lai et al. [15] introduced conditional diagnosability under the assumption that all the neighbors of any processor in a multiprocessor system cannot be faulty at the same time. The conditional diagnosability of interconnection networks has been investigated (see [4, 6, 7, 8, 11, 26, 27, 28, 31, 32, 33]).

In 2012, Peng et al. proposed $g$-good-neighbor conditional diagnosability [20], which extended the concept of conditional diagnosability. This requires that every fault-free vertex has at least $g$ fault-free neighbors. Peng et al. [20] studied the $g$-good-neighbor conditional diagnosability of the $n$-dimensional hypercube $Q_{n}$ under the PMC model. Since then, many researchers have studied this topic (see [14, 18, 23, 24, 25, 29, 30]).

The $(n,k)$-star network $S_{n,k}$, proposed by Chiang and Chen [3], is an extension of the $n$-dimensional star graph $S_{n}$. The network $S_{n,k}$ preserves many ideal properties of $S_{n}$. In recent years, $S_{n,k}$ has received considerable attention [4, 7, 8, 9, 13, 16, 17, 25, 31]. In particular, Xu et al. [25] derived the following result about the $g$-good-neighbor conditional diagnosability of $S_{n,k}$ under the PMC model and the MM∗ model.

Theorem 1.1** (Xu et al. [25])**

The $g$-good-neighbor conditional diagnosabilities of the $(n,k)$-star graph $S_{n,k}$ under the PMC model and the MM∗ model are

[TABLE]

and

[TABLE]

respectively.

However, there are some unknown cases (see Table 1).

In this paper, we determine the $g$-good-neighbor conditional diagnosability of $(n,k)$-star networks $S_{n,k}$ for all the remaining cases (see Table 2). Recently, Li et al. [14] determined the $g$-good-neighbor conditional diagnosability of the star graph $S_{n}$ under the PMC model and the MM∗ model as follows.

Theorem 1.2** (Li et al. [14])**

The $g$-good-neighbor conditional diagnosabilities of the star graph $S_{n}$ with $n\geq 4$ for $0\leq g\leq n-2$ under the PMC model and the MM∗ model are $(n-g)(g+1)!-1$.

Note that $S_{n,n-1}$ is isomorphic to $S_{n}$. Thus, the results in Table 2 extend their results when $1\leq g\leq n-2$.

The rest of this paper is organized as follows. Section 2 introduces some terminology and preliminaries. Our main results are given in Section 3. Finally, Section 4 concludes the paper.

2 Terminology and preliminaries

An undirected simple graph G=\big{(}V(G),E(G)\big{)} is used to represent a system (or a network) where each vertex represents a processor and each edge represents a link. A subgraph $H$ of $G$ is a graph with $V(H)\subseteq V(G)$, $E(H)\subseteq E(G)$, and the endpoints of every edge in $E(H)$ belonging to $V(H)$. For an arbitrary subset $F\subseteq V(G)$, we use $G-F$ to denote the graph obtained by removing all the vertices in $F$ from $G$. Given a nonempty vertex subset $V^{\prime}$ of $V(G)$, the induced subgraph by $V^{\prime}$ in $G$, denoted by $G[V^{\prime}]$, is a graph in which the vertex set is $V^{\prime}$ and the edge set is the set of all the edges of $G$ with both endpoints in $V^{\prime}$. For a given vertex $v$, we define the neighborhood $N_{G}(v)$ of $v$ in $G$ to be the set of vertices adjacent to $v$. The degree of vertex $v$, denoted by $d_{G}(v)$, is the number of vertices in $N_{G}(v)$. The minimum degree of a graph $G$, denoted by $\delta(G)$, is $\min_{v\in V(G)}d_{G}(v)$. A graph $G$ is $k$-regular if $d_{G}(v)=k$ for any $v\in V$. For a given set $A\subseteq G$, we denote by $N_{G}(A)$ the set \big{(}\bigcup_{v\in V(A)}N_{G}(v)\big{)}-V(A). For neighborhoods and degrees, we omit the subscripts of the graphs when no confusion arises. The symmetric difference of two sets $F_{1}$ and $F_{2}$ is defined as the set $F_{1}\bigtriangleup F_{2}=(F_{1}-F_{2})\cup(F_{2}-F_{1})$. Please refer to [2] for graph-theoretical terminology and notation undefined here.

Now we focus on star graphs $S_{n}$ and $(n,k)$-star networks $S_{n,k}$. For a given integer $n$ with $n\geq 1$, we set $I_{n}=\{1,2,\ldots,n\}$ and $I^{\prime}_{n}=I_{n}-\{1\}$. Let $P(n,k)=\{p_{1}p_{2}\ldots p_{k}\mid p_{i}\in I_{n},p_{i}\neq p_{j},1\leq i\neq j\leq k\}$, the set of $k$-arrangements on $I_{n}$, where $k\in I_{n}$. We will abbreviate $P(n,n)$ as $P(n)$.

Definition 2.1** (Akers and Krishnamurthy [1])**

An $n$-dimensional star graph $S_{n}$ is a graph with vertex set $P(n)$, a vertex $p=p_{1}p_{2}\ldots p_{i}\ldots p_{n}$ being linked a vertex $q$ if and only if $q=p_{i}p_{2}\ldots p_{i-1}p_{1}p_{i+1}\ldots p_{n}$ for some $i\in I^{\prime}_{n}$ (See Figure 1).

Definition 2.2** (Chiang and Chen [3])**

An $(n,k)$-star graph $S_{n,k}$ (See Figure 2) is a graph with vertex set $P(n,k)$, a vertex $p=p_{1}p_{2}\ldots p_{i}\ldots p_{k}$ being linked a vertex $q$ if and only if $q$ is:

1. (a)

$p_{i}p_{2}\ldots p_{i-1}p_{1}p_{i+1}\ldots p_{k}$, where $i\in I^{\prime}_{k}$ (swap $p_{1}$ with $p_{i}$); or 2. (b)

$p^{\prime}_{1}p_{2}p_{3}\ldots p_{k}$, where $p^{\prime}_{1}\in I_{n}-\{p_{i}\mid i\in I_{k}\}$ (replace $p_{1}$ by $p^{\prime}_{1}$).

By the definition, the $(n,k)$-star graph is an $(n-1)$-regular, $(n-1)$-connected graph and $S_{n,n-1}$ is isomorphic to $S_{n}$ (see [3]).

Now we introduce two models for fault diagnosis.

In the PMC model, all processors in the system under diagnosis can test one another. The set of tests can be represented by a directed graph $G=(V,E)$, in which each vertex represents a processor, and an edge $(u,v)$ indicates that the processor $u$ has tested processor $v$. The outcome of vertex $u$ testing vertex $v$ is denoted by $\sigma(u,v)$, where

[TABLE]

where $F$ is the set of faulty vertices.

In the MM∗ model, a processor executes comparisons for any pair of its neighboring processors. A graph $G=(V,E)$ is used to represent a system, where each vertex represents a processor and each edge represents a link. Assign a task to each vertex. The vertex $w$ is a comparator of a pair of vertices $\{u,v\}$ if $(u,w)\in E$ and $(v,w)\in E$. The outcome of this comparison is denoted by \sigma\big{(}(u,v)_{w}\big{)}, where

[TABLE]

where $F$ is the set of faulty vertices.

The collection of all outcomes is called a syndrome $\sigma$. The diagnosis problem involves using the syndrome to determine the status (faulty or fault free) of each processor in the system. For a given syndrome $\sigma$, a subset $F\subseteq V$ is said to be consistent with $\sigma$ if the syndrome $\sigma$ can be produced from the faulty set $F$. In the PMC model, $F$ is said to be consistent with $\sigma$ if the syndrome $\sigma$ can be produced from the situation that, for any $(u,v)\in E$ such that $u\notin F$, $\sigma(u,v)=1$ if and only if $v\in F$. In the MM∗ model, $F$ is said to be consistent with $\sigma$ if the syndrome $\sigma$ can be produced from the situation that, for any $(u,w)\in E$ and $(v,w)\in E$ such that $w\notin F$, \sigma\big{(}(u,v)_{w}\big{)}=1 if and only if $\{u,v\}\cap F\neq\emptyset$. Therefore, on the one hand, a faulty set $F$ may produce a number of different syndromes. On the other hand, different faulty sets may produce the same syndrome. Define $\sigma(F)=\{\sigma\mid F~{}{\rm is~{}consistent~{}with}~{}\sigma\}$. Two distinct sets $F_{1}$, $F_{2}\subseteq V$ are said to be indistinguishable if $\sigma(F_{1})\cap\sigma(F_{2})\neq\emptyset$; otherwise, $F_{1}$ and $F_{2}$ are said to be distinguishable. We say that $(F_{1},F_{2})$ is an indistinguishable pair if $\sigma(F_{1})\cap\sigma(F_{2})\neq\emptyset$; otherwise, $(F_{1},F_{2})$ is a distinguishable pair.

The following lemmas give necessary and sufficient conditions for a pair of sets to be distinguishable under the PMC model and the MM∗ model.

Lemma 2.3** (Dahbura and Masson [10])**

Let $G=(V,E)$ be a graph. For any two distinct sets $F_{1},F_{2}\subseteq V$, $(F_{1},F_{2})$ is a distinguishable pair under the PMC model if and only if there exists a vertex $u\in V-(F_{1}\cup F_{2})$ and there exists a vertex $v\in F_{1}\bigtriangleup F_{2}$ such that $(u,v)\in E$ (See Figure 3).

Lemma 2.4** (Sengupta and Dahbura [22])**

Let $G=(V,E)$ be a graph. For any two distinct sets $F_{1},F_{2}\subseteq V$, $F_{1}$ and $F_{2}$ are distinguishable under the MM∗ model if and only if any one of the following conditions is satisfied (See Figure 4).

1. (1)

There are two vertices $u,w\in V-(F_{1}\cup F_{2})$ and there is a vertex $v\in F_{1}\bigtriangleup F_{2}$ such that $(u,v)\in E$ and $(u,w)\in E$. 2. (2)

There are two vertices $u,v\in F_{1}-F_{2}$ and there is a vertex $w\in V-(F_{1}\cup F_{2})$ such that $(u,w)\in E$ and $(v,w)\in E$. 3. (3)

There are two vertices $u,v\in F_{2}-F_{1}$ and there is a vertex $w\in V-(F_{1}\cup F_{2})$ such that $(u,w)\in E$ and $(v,w)\in E$.

Next, we introduce the diagnosability, conditional diagnosability, $g$-good-neighbor conditional diagnosability, and $R_{g}$-connectivity of a graph in the following statements.

Definition 2.5** (Dahbura and Masson [10])**

For a graph $G=(V,E)$, $G$ is $t$-diagnosable if all faulty processors can be detected without replacement, provided that the number of faults does not exceed $t$. The diagnosability $t(G)$ of graph $G$ is the maximum value of $t$ such that $G$ is $t$-diagnosable.

The diagnosability of multiprocessor systems, as defined above, assumes that all neighbors of any processor may fail simultaneously. However, the probability that all the neighbors of a processor fail is very small. In 2005, Lai et al. [15] introduced conditional diagnosability under the assumption that all the neighbors of any processor in a multiprocessor system cannot be faulty at the same time.

Definition 2.6** (Lai et al. [15])**

For a graph $G=(V,E)$, $G$ is conditionally $t$-diagnosable if $G$ is $t$-diagnosable, provided that for any processor $v\in V$, the set of faults does not contain the neighborhood $N(v)$ as a subset. The conditional diagnosability $t_{c}(G)$ of graph $G$ is the maximum value of $t$ such that $G$ is conditionally $t$-diagnosable.

Inspired by the concept of conditional diagnosability, Peng et al. [20] proposed $g$-good-neighbor conditional diagnosability in 2012, which extended the concept of conditional diagnosability.

Definition 2.7** (Peng et al. [20])**

For a graph $G=(V,E)$, a faulty set $F\subseteq V$ is called a $g$-good-neighbor conditional faulty set if $|N(v)\cap(V-F)|\geq g$ for each node $v$ in $V-F$. A graph $G$ is $g$-good-neighbor conditional $t$-diagnosable if $G$ is $t$-diagnosable, provided that every faulty set is a $g$-good-neighbor conditional faulty set. The $g$-good-neighbor conditional diagnosability $t_{g}(G)$ of $G$ is the maximum value of $t$ such that $G$ is $g$-good-neighbor conditionally $t$-diagnosable.

Definition 2.8** (Yuan et al. [29])**

A $g$-good-neighbor conditional cut of a graph $G$ is a $g$-good-neighbor conditional faulty set $F$ such that $G-F$ is disconnected. The minimum cardinality of $g$-good-neighbor cuts is said to be the $R_{g}$-connectivity of $G$, denoted by $\kappa^{g}(G)$.

The same concepts as $g$-good-neighbor conditional cut and $R_{g}$-connectivity can be found in [13]. We restate their definitions as follows.

• •

Let $G$ be a connected graph. A subset $T\subset V(G)$, if any, is called an $h$-vertex-cut if $G-T$ is disconnected and has the minimum degree at least $h$. The $h$-super connectivity $\kappa^{(h)}_{s}(G)$ of $G$ is defined as the minimum cardinality over all $h$-vertex-cuts of $G$.

In this paper, we adopt the notation $\kappa^{g}(G)$.

The $R_{g}$-connectivity of an $(n,k)$-star graph $S_{n,k}$ was determined as follows.

Lemma 2.9** (Li et al. [13])**

For $2\leq k\leq n-1$ and $n-k\leq g\leq n-2$, $\kappa^{g}(S_{n,k})=\dfrac{(g+1)!(n-g-1)}{(n-k)!}$.

The following lemmas give necessary and sufficient conditions for a system to be $g$-good-neighbor $t$-diagnosable under the PMC model and under the MM∗ model.

Lemma 2.10** (Peng et al. [20] and Dahbura and Masson [10])**

A graph $G=(V,E)$ is $g$-good-neighbor $t$-diagnosable under the PMC model if and only if there is an edge $(u,v)\in E$ with $u\in V-(F_{1}\cup F_{2})$ and $v\in F_{1}\bigtriangleup F_{2}$ for each distinct pair of $g$-good-neighbor conditional faulty sets $F_{1}$ and $F_{2}$ of $V$ with $|F_{1}|\leq t$ and $|F_{2}|\leq t$ (See Figure 3).

Lemma 2.11** (Sengupta and Dahbura [22] and Yuan et al. [29])**

A graph $G=(V,E)$ is $g$-good-neighbor $t$-diagnosable under the MM∗ model if and only if each distinct pair of $g$-good-neighbor conditional faulty sets $F_{1}$ and $F_{2}$ of $V$ with $|F_{1}|\leq t$ and $|F_{2}|\leq t$ satisfies one of the following conditions (See Figure 4).

1. (1)

There are two vertices $u,w\in V-(F_{1}\cup F_{2})$ and there is a vertex $v\in F_{1}\bigtriangleup F_{2}$ such that $(u,v)\in E$ and $(u,w)\in E$. 2. (2)

There are two vertices $u,v\in F_{1}-F_{2}$ and there is a vertex $w\in V-(F_{1}\cup F_{2})$ such that $(u,w)\in E$ and $(v,w)\in E$. 3. (3)

There are two vertices $u,v\in F_{2}-F_{1}$ and there is a vertex $w\in V-(F_{1}\cup F_{2})$ such that $(u,w)\in E$ and $(v,w)\in E$.

Next, we introduce the split-graph, proposed in [13], which will be used in the proof of our main results.

Definition 2.12** (Li et al. [13])**

Let $G$ be a graph and $t$ be a positive integer. A $t$-split graph $G^{t}$ of $G$ is a graph obtained from $G$ by replacing each vertex $x$ by a set $V_{x}$ of $t$ independent vertices, and replacing each edge $e=xy$ by a perfect matching $E_{e}$ between $V_{x}$ and $V_{y}$ (See Figure 5).

Li et al. [13] obtained the relationship between $S_{n}$ and $S_{n,k}$ as follows.

Lemma 2.13** (Li et al. [13])**

For any $k$ with $2\leq k\leq n-1$, there is an $(n-k)!$-split graph of $S_{n,k}$ that is isomorphic to a star graph $S_{n}$.

The following lemma is very useful in proving our main results.

Lemma 2.14** (Li and Lu [14])**

Let $H$ be a subgraph of $S_{n}$ for $n\geq 2$ and $\delta(H)\geq g$, where $g\in I_{n-1}$. Then $|V(H)|\geq(g+1)!$.

3 Main Results

Lemma 3.1

Let $H$ be a subgraph of $S_{n,k}$ for $2\leq k\leq n-1$. If $\delta(H)\geq g$, where $g\in I_{n-1}$, then $|V(H)|\geq\dfrac{(g+1)!}{(n-k)!}$.

*Proof. *Suppose that $H$ is a subgraph of $S_{n,k}$. Since $2\leq k\leq n-1$, by Lemma 2.13, there is an $(n-k)!$-split graph of $S_{n,k}$, denoted by $S_{n,k}^{(n-k)!}$ which is isomorphic to a star graph $S_{n}$. Let $H^{(n-k)!}$ be an $(n-k)!$-split graph of $H$ satisfying that it is also a subgraph of $S_{n,k}^{(n-k)!}$. Thus, $|V(H^{(n-k)!})|=(n-k)!|V(H)|$. Since $\delta(H)\geq g$, we have $\delta(H^{(n-k)!})\geq g$. By Lemma 2.14, $|V(H^{(n-k)!})|\geq(g+1)!$. Therefore, we have $|V(H)|\geq\dfrac{(g+1)!}{(n-k)!}$.

We obtain the desired result. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Note that $S_{n,k}$ is $(n-1)$-regular. Then $t_{n-1}(S_{n,k})=0$ under the PMC model and the MM∗ model. In the following, we assume that $g\leq n-2$.

Now, we consider the upper bound of the $g$-good-neighbor conditional diagnosability of $(n,k)$-star network $S_{n,k}$ for $2\leq k\leq n-1$ and $n-k\leq g\leq n-2$ under the PMC model and the MM∗ model.

Lemma 3.2

For $2\leq k\leq n-1$ and $n-k\leq g\leq n-2$, we have $t_{g}(S_{n,k})\leq\dfrac{(g+1)!(n-g)}{(n-k)!}-1$ under the PMC model and the MM∗ model.

*Proof. *Let

[TABLE]

$F_{1}=N(A)$ and $F_{2}=F_{1}\cup A$ (see Figure 6). Then $|A|=\dfrac{(g+1)!}{(n-k)!}$. For $F_{1}=N(A)$, we have

[TABLE]

Choose a vertex $u=p_{1}p_{2}\ldots p_{k-(n-g-1)}12\ldots(n-g-1)\in A$. By Definition 2.2, we have

[TABLE]

[TABLE]

Then $|N(u)\cap A|=g$ and $|N(u)\cap F_{1}|=n-g-1$. By the definition of $S_{n,k}$, no two vertices in $A$ share a common neighbor in $F_{1}$. It follows that $|F_{1}|=|A|(n-g-1)=\dfrac{(g+1)!(n-g-1)}{(n-k)!}$ and $|F_{2}|=|F_{1}|+|A|=\dfrac{(g+1)!(n-g)}{(n-k)!}$. Note that $A=F_{1}\bigtriangleup F_{2}$ and $F_{1}=N(A)$. Since there is no edge between $F_{1}\bigtriangleup F_{2}$ and $V(S_{n,k})-(F_{1}\cup F_{2})$, by Lemmas 2.3 and 2.4, we conclude that $F_{1}$ and $F_{2}$ are indistinguishable under the PMC model and the MM∗ model.

Now, we verify that both $F_{1}$ and $F_{2}$ are $g$-good-neighbor conditional faulty sets.

Suppose $u\in V(S_{n,k})-F_{2}$. If $u\notin N(F_{2})$, then $N(u)\subseteq V(S_{n,k})-F_{2}$ and $|N(u)|=n-1>g$. If $u\in N(F_{2})$, then there exists $v\in F_{1}$ such that $uv\in E(S_{n,k})$. We can assume that $v=ip_{2}\ldots p_{k-(n-g-1)}1\ldots(i-1)p_{1}(i+1)\ldots(n-g-1)$ for some fixed $i\in I_{n-g-1}$. Thus, $u$ has three forms, i.e.,

[TABLE]

[TABLE]

[TABLE]

No matter which form $u$ has, $N(u)\cap F_{1}=\{v\}$. Since $S_{n,k}$ is $(n-1)$-regular, $|N(u)-F_{2}|=(n-1)-1\geq g$. Thus, $F_{2}$ is a $g$-good-neighbor conditional faulty set.

Suppose $u\in V(S_{n,k})-F_{1}$. If $u\in V(LTQ_{n})-F_{2}$, then we obtain the desired result by the same proof as above. If $u\notin V(LTQ_{n})-F_{2}$, then $u\in A$. Note that $S_{n,k}[A]$ is $g$-regular. Thus, $|N(u)\cap A|=g$. So $F_{1}$ is a $g$-good-neighbor conditional faulty set.

By Lemmas 2.10 and 2.11, we have $t_{g}(S_{n,k})\leq\dfrac{(g+1)!(n-g)}{(n-k)!}-1$ under the PMC model and the MM∗ model. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Next, we consider the lower bound of the $g$-good-neighbor conditional diagnosability of $(n,k)$-star network $S_{n,k}$ for $2\leq k\leq n-1$ and $n-k\leq g\leq n-2$ under the PMC model and the MM∗ model, respectively.

Lemma 3.3

For $2\leq k\leq n-1$ and $n-k\leq g\leq n-2$ with $n\geq 4$, we have $t_{g}(S_{n,k})\geq\dfrac{(g+1)!(n-g)}{(n-k)!}-1$ under the PMC model.

*Proof. *Suppose that $F_{1}$ and $F_{2}$ are any two distinct $g$-good-neighbor conditional faulty sets and they are indistinguishable. We will prove the lemma by showing that $|F_{1}|\geq\dfrac{(g+1)!(n-g)}{(n-k)!}$ or $|F_{2}|\geq\dfrac{(g+1)!(n-g)}{(n-k)!}$.

If $V(S_{n,k})=F_{1}\cup F_{2}$, then $|F_{1}|\geq\dfrac{1}{2}|V(S_{n,k})|\geq\dfrac{n!}{2(n-k)!}\geq\dfrac{(g+1)!(n-g)}{(n-k)!}$ or $|F_{2}|\geq\dfrac{1}{2}|V(S_{n,k})|\geq\dfrac{n!}{2(n-k)!}\geq\dfrac{(g+1)!(n-g)}{(n-k)!}$ for $n\geq 4$.

Now, we suppose $V(S_{n,k})\neq F_{1}\cup F_{2}$. Since $F_{1}$ and $F_{2}$ are indistinguishable, there are no edges between $V(S_{n,k})-(F_{1}\cup F_{2})$ and $F_{1}\bigtriangleup F_{2}$ by Lemma 2.3. Thus, $S_{n,k}-(F_{1}\cap F_{2})$ is disconnected. Without loss of generality, we assume that $F_{2}-F_{1}\neq\emptyset$. Note that $F_{1}$ and $F_{2}$ are both $g$-good-neighbor conditional faulty sets. Then $\delta(S_{n,k}[V(S_{n,k})-(F_{1}\cup F_{2})])\geq g$ and $\delta(S_{n,k}[F_{2}-F_{1}])\geq g$. Thus, we have $|F_{2}-F_{1}|\geq\dfrac{(g+1)!}{(n-k)!}$ by Lemma 3.1. If $F_{1}-F_{2}\neq\emptyset$, then $\delta(S_{n,k}[F_{1}-F_{2}])\geq g$. Therefore, $F_{1}\cap F_{2}$ is a $g$-good-neighbor conditional cut of $S_{n,k}$. By Lemma 2.9, we obtain that $|F_{1}\cap F_{2}|\geq\dfrac{(g+1)!(n-g-1)}{(n-k)!}$. Hence, $|F_{2}|=|F_{2}-F_{1}|+|F_{1}\cap F_{2}|\geq\dfrac{(g+1)!}{(n-k)!}+\dfrac{(g+1)!(n-g-1)}{(n-k)!}=\dfrac{(g+1)!(n-g)}{(n-k)!}$.

This completes the proof of Lemma 3.3. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Lemma 3.4

For $2\leq k\leq n-1$ and $n-k\leq g\leq n-2$ with $n\geq 4$, we have $t_{g}(S_{n,k})\geq\dfrac{(g+1)!(n-g)}{(n-k)!}-1$ under the MM∗ model.

*Proof. *If $g=1$, then $k=n-1$. Note that $S_{n,n-1}$ is isomorphic to $S_{n}$. By Theorems 1.1 and 1.2, $t_{1}(S_{n,n-1})=2n-3$ for $n\geq 4$. The result holds.

Now assume $g\geq 2$. Suppose that $F_{1}$ and $F_{2}$ are any two distinct $g$-good-neighbor conditional faulty sets and they are indistinguishable. We will prove the lemma by showing that $|F_{1}|\geq\dfrac{(g+1)!(n-g)}{(n-k)!}$ or $|F_{2}|\geq\dfrac{(g+1)!(n-g)}{(n-k)!}$.

If $V(S_{n,k})=F_{1}\cup F_{2}$, then $|F_{1}|\geq\dfrac{1}{2}|V(S_{n,k})|\geq\dfrac{n!}{2(n-k)!}\geq\dfrac{(g+1)!(n-g)}{(n-k)!}$ or $|F_{2}|\geq\dfrac{1}{2}|V(S_{n,k})|\geq\dfrac{n!}{2(n-k)!}\geq\dfrac{(g+1)!(n-g)}{(n-k)!}$ for $n\geq 4$.

Now, we suppose $V(S_{n,k})\neq F_{1}\cup F_{2}$. Without loss of generality, we assume that $F_{2}-F_{1}\neq\emptyset$. We shall show that there is no edge between $F_{1}\bigtriangleup F_{2}$ and $V(S_{n,k})-(F_{1}\cup F_{2})$. Otherwise, there exists an edge $uv\in E(S_{n,k})$, where $u\in F_{1}\bigtriangleup F_{2}$ and $v\in V(S_{n,k})-(F_{1}\cup F_{2})$. Without loss of generality, we can assume that $u\in F_{2}-F_{1}$. Since $F_{1}$ is a $g$-good-neighbor conditional faulty set with $g\geq 2$, $v$ has at least two neighbors in $S_{n,k}-F_{1}$. Thus, $v$ has a neighbor $w~{}(w\neq u)$ in $F_{2}-F_{1}$ or $V(S_{n,k})-(F_{1}\cup F_{2})$, which contradicts Lemma 2.4. By the same discussion as Lemma 3.3, we complete the proof of this lemma. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Now, we get the result below by Lemmas 3.2, 3.3, and 3.4 directly.

Theorem 3.5

Let $S_{n,k}$ be $(n,k)$-star networks with $2\leq k\leq n-1$ and $n\geq 4$. Then the $g$-good-neighbor conditional diagnosability of $S_{n,k}$ under the PMC model and the MM∗ model are both $t_{g}(S_{n,k})=\dfrac{(g+1)!(n-g)}{(n-k)!}-1$ for $n-k\leq g\leq n-2$.

Now we discuss the other unknown cases in Table 1.

Theorem 3.6

The $g$-good-neighbor conditional diagnosability of $S_{n,1}$ with $n\geq 4$ under the PMC model and the MM∗ model are both $t_{g}(S_{n,1})=n-g-1$ for $\left\lfloor\dfrac{n}{2}\right\rfloor\leq g\leq n-2$. What is more, the $1$-good-neighbor conditional diagnosabilities of $S_{3,1}$ under the PMC model and the MM∗ model are $1$ and [math], respectively.

*Proof. *When $k=1$, $S_{n,1}$ is isomorphic to $K_{n}$ which is a complete graph.

First, we consider the $1$-good-neighbor conditional diagnosability of $S_{3,1}$ under the PMC model and the MM∗ model.

Note that any $1$-good-neighbor conditional faulty set of $S_{3,1}$ contains at most one vertex. We have $t_{1}(S_{3,1})\leq 1$. Let $F_{1}$ and $F_{2}$ are any two distinct $1$-good-neighbor conditional faulty sets of $S_{3,1}$ with $|F_{1}|\leq 1$ and $|F_{2}|\leq 1$. Since $V(S_{3,1})-(F_{1}\cup F_{2})\neq\emptyset$, there exists one edge between $F_{1}\bigtriangleup F_{2}$ and $V(S_{n,k})-(F_{1}\cup F_{2})$. By Lemma 2.3, we know $F_{1}$ and $F_{2}$ are distinguishable under the PMC model. Thus, $t_{1}(S_{3,1})=1$ under the PMC model.

Suppose $V(S_{3,1})=\{u_{1},u_{2},u_{3}\}$. Let $F_{1}=\{u_{1}\}$ and $F_{2}=\{u_{2}\}$. Then $F_{1}\bigtriangleup F_{2}=\{u_{1},u_{2}\}$ and $V(S_{3,1})-(F_{1}\cup F_{2})=\{u_{3}\}$. Obviously, $F_{1}$ and $F_{2}$ are $1$-good-neighbor conditional faulty sets of $S_{3,1}$. By Lemma 2.4, $F_{1}$ and $F_{2}$ are indistinguishable under the MM∗ model. Hence, $t_{1}(S_{3,1})=0$ under the MM∗ model.

Next, we consider the $g$-good-neighbor conditional diagnosability of $S_{n,1}$ with $n\geq 4$ for $\left\lfloor\dfrac{n}{2}\right\rfloor\leq g\leq n-2$ under the PMC model and the MM∗ model.

Note that the $g$-good-neighbor conditional faulty set of $S_{n,1}$ contains at most $n-g-1$ vertices. We have $t_{g}(S_{n,1})\leq n-g-1$ under the PMC model and the MM∗ model.

To proceed, we show that $t_{g}(S_{n,1})\geq n-g-1$ under the PMC model and the MM∗ model.

Suppose that $F_{1}$ and $F_{2}$ are any two distinct $g$-good-neighbor conditional faulty sets and $|F_{i}|\leq n-g-1$ for each $i\in I_{2}$. We will show that they are distinguishable. Without loss of generality, we assume that $F_{2}-F_{1}\neq\emptyset$.

If $V(S_{n,1})=F_{1}\cup F_{2}$, then $n=|V(S_{n,1})|=|F_{1}\cup F_{2}|=|F_{1}|+|F_{2}|-|F_{1}\cap F_{2}|\leq|F_{1}|+|F_{2}|\leq 2(n-g-1)\leq 2(n-\left\lfloor\dfrac{n}{2}\right\rfloor-1)<n$, which is a contradiction.

Now, we suppose $V(S_{n,1})\neq F_{1}\cup F_{2}$. We have

[TABLE]

Note that $S_{n,1}$ is isomorphic to $K_{n}$. Under the PMC model, by Lemma 2.3, we know that $F_{1}$ and $F_{2}$ are distinguishable. Thus, we have $t_{g}(S_{n,1})\geq n-g-1$ under the PMC model.

If $|V(S_{n,1})-(F_{1}\cup F_{2})|\geq 2$, then by Lemma 2.4(1), we know that $F_{1}$ and $F_{2}$ are distinguishable. See Figure 7(a).

If $|V(S_{n,1})-(F_{1}\cup F_{2})|=1$, then by (1), we have $n$ is odd and $F_{1}\cap F_{2}=\emptyset$. We can also know that $|F_{1}|=|F_{2}|=\left\lfloor\dfrac{n}{2}\right\rfloor\geq 2$. Thus, by Lemma 2.4(3), we know that $F_{1}$ and $F_{2}$ are distinguishable. See Figure 7(b).

Thus, we have $t_{g}(S_{n,1})\geq n-g-1$ under the MM∗ model.

As mentioned above, we obtain the desired result. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Finally, we discuss $t_{1}(S_{n,2})$ under the MM∗ model.

Theorem 3.7

Under the MM∗ model, we have

[TABLE]

*Proof. *First, we consider $t_{1}(S_{3,2})$ under the MM∗ model. Note that $S_{3,2}$ is a cycle with six vertices. Suppose $V(S_{3,2})=\{u_{1},u_{2},\ldots,u_{6}\}$ and $E(S_{3,2})=\{u_{1}u_{2},u_{2}u_{3},u_{3}u_{4},u_{4}u_{5},u_{5}u_{6},u_{1}u_{6}\}$.

Let $F_{1}=\{u_{1},u_{2}\}$ and $F_{2}=\{u_{4},u_{5}\}$. Obviously, $F_{1}$ and $F_{2}$ are $1$-good-neighbor conditional faulty sets of $S_{3,2}$. By Lemma 2.4, $F_{1}$ and $F_{2}$ are indistinguishable under the MM∗ model. See Figure 8. Thus, we have $t_{1}(S_{3,2})\leq 1$ under the MM∗ model.

On the other hand, by transitivity of $S_{3,2}$, we may suppose that $F_{1}=\{u_{1}\}$ and $F_{2}=\{u_{j}\}$ (or $F_{2}=\emptyset$) are any two distinct $1$-good-neighbor conditional faulty sets of $S_{3,2}$.

When $F_{2}=\{u_{j}\}$, we assume that $2\leq j\leq 6$.

If $4\leq j\leq 6$, then $u_{2},u_{3}\in V(S_{3,2})-(F_{1}\cup F_{2})$. Since $u_{1}\in F_{1}\bigtriangleup F_{2}$ and $u_{1}u_{2},u_{2}u_{3}\in E(S_{3,2})$, we have that $F_{1}$ and $F_{2}$ are distinguishable under the MM∗ model. See Figure 9 for $j=4$.

If $2\leq j\leq 3$, then $u_{5},u_{6}\in V(S_{3,2})-(F_{1}\cup F_{2})$. Since $u_{1}\in F_{1}\bigtriangleup F_{2}$ and $u_{1}u_{6},u_{6}u_{5}\in E(S_{3,2})$, we have that $F_{1}$ and $F_{2}$ are distinguishable under the MM∗ model. See Figure 9 for $j=3$.

When $F_{2}=\emptyset$, the result also holds.

By Lemma 2.11, we have $t_{1}(S_{3,2})\geq 1$ under the MM∗ model. Therefore, $t_{1}(S_{3,2})=1$ under the MM∗ model.

Now, we consider $t_{1}(S_{n,2})$ under the MM∗ model, where $n\geq 4$. Our discussion is divided into two steps.

Step 1: Show that $t_{1}(S_{n,2})\geq n-1$ under the MM∗ model, where $n\geq 4$.

Suppose that $F_{1}$ and $F_{2}$ are any two distinct $1$-good-neighbor conditional faulty sets and they are indistinguishable. We will show that $|F_{1}|\geq n$ or $|F_{2}|\geq n$.

If $V(S_{n,2})=F_{1}\cup F_{2}$, then $|F_{1}|\geq\dfrac{1}{2}|V(S_{n,2})|=\dfrac{n(n-1)}{2}\geq n$ or $|F_{2}|\geq\dfrac{1}{2}|V(S_{n,2})|=\dfrac{n(n-1)}{2}\geq n$ for $n\geq 4$.

Now, we suppose $V(S_{n,2})\neq F_{1}\cup F_{2}$. Without loss of generality, we assume that $F_{2}-F_{1}\neq\emptyset$.

If $|F_{1}\cap F_{2}|\geq n-1$, then $|F_{2}|=|F_{2}-F_{1}|+|F_{1}\cap F_{2}|\geq 1+(n-1)=n$.

Now, we suppose $|F_{1}\cap F_{2}|\leq n-2$. Let $W_{1},\ldots,W_{c}$ be the components of $S_{n,2}-(F_{1}\cup F_{2})$ such that $|V(W_{1})|\leq\ldots\leq|V(W_{c})|$, where $c\geq 1$. For any component $W_{i}$, if $|V(W_{i})|\geq 2$, then there is no edge between $W_{i}$ and $F_{1}\bigtriangleup F_{2}$. Otherwise, it contradicts the fact that $F_{1}$ and $F_{2}$ are indistinguishable by Lemma 2.4.

If $|V(W_{c})|\geq 2$, then $F_{1}\cap F_{2}$ is a cut of $S_{n,2}$. Since the connectivity of $S_{n,2}$ is $n-1$, we have $|F_{1}\cap F_{2}|\geq n-1$, which contradicts the assumption that $|F_{1}\cap F_{2}|\leq n-2$.

Next, we assume that $|V(W_{c})|=1$. If $F_{1}-F_{2}=\emptyset$, then $S_{n,2}-F_{1}-F_{2}=S_{n,2}-F_{2}$. The vertex in $W_{1}$ is one isolated vertex in $S_{n,k}-F_{2}$, which contradicts the fact that $F_{2}$ is a $1$-good neighbor conditional faulty set. We suppose $F_{1}-F_{2}\neq\emptyset$. Let $W=\bigcup_{1\leq i\leq c}V(W_{i})$. Then $V(S_{n,2})=W\cup(F_{1}\cup F_{2})$. Arbitrarily choose a vertex $w\in W$. Then, $N(w)\subseteq F_{1}\cup F_{2}$. Since $F_{1}$ and $F_{2}$ are indistinguishable, $|N(w)\cap(F_{2}-F_{1})|\leq 1$ and $|N(w)\cap(F_{1}-F_{2})|\leq 1$ by Lemma 2.4. Owing to the fact that $F_{1}$ and $F_{2}$ are $1$-good-neighbor conditional faulty sets, we have $|N(w)\cap(F_{2}-F_{1})|=|N(w)\cap(F_{1}-F_{2})|=1$ and $|N(w)\cap(F_{1}\cap F_{2})|=n-3\leq|F_{1}\cap F_{2}|$. Thus,

[TABLE]

It follows that $|W|\leq\dfrac{(n-2)(n-1)}{n-3}=n+\dfrac{2}{n-3}\leq n+2$ when $n\geq 4$. Thus,

[TABLE]

Therefore, for $n\geq 4$, we have

[TABLE]

Based on the above discussion, we obtain the desired result.

Step 2: Show that $t_{1}(S_{n,2})\leq n-1$ under the MM∗ model, where $n\geq 4$.

By using the construction of Chang et al. [4], let

[TABLE]

[TABLE]

and

[TABLE]

Then $|A|=(n-4)+3=n-1$ and $|F_{1}|=|F_{2}|=n$. See Figure 10.

We conclude that $F_{1}$ and $F_{2}$ are indistinguishable $1$-good-neighbor conditional faulty sets under the MM∗ model from the proof of [4]. By Lemma 2.11, we have $t_{1}(S_{n,2})\leq n-1$ under the MM∗ model, where $n\geq 4$.

By Steps 1 and 2, we conclude that $t_{1}(S_{n,2})=n-1$ under the MM∗ model, where $n\geq 4$.

This completes the proof of Theorem 3.7. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

4 Conclusions

In this paper, we determined the $g$-good-neighbor conditional diagnosability of $(n,k)$-star networks $S_{n,k}$ for all the remaining cases with $1\leq k\leq n-1$ and $n\geq 3$ for $1\leq g\leq n-k$ under the PMC model and the MM∗ model (see Table 2). Future research on this topic will involve studying the $g$-good-neighbor conditional diagnosability of many network topologies.