Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks THLNs

TL;DR

This paper investigates the fault-tolerant path embedding in twisted hypercube-like networks, demonstrating the existence of fault-free paths of various lengths between any two correct vertices under certain fault conditions.

Contribution

It establishes new bounds for fault-tolerant path embedding in $THLNs$, considering different vertex pair types and fault scenarios, advancing network reliability analysis.

Findings

Existence of fault-free paths of specified lengths between any two correct vertices.

Path length bounds depend on vertex pair type and fault set size.

Applicable for networks with dimension n ≥ 5.

Abstract

The twisted hypercube-like networks($THLNs$) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of $n$-dimensional($n$-$D$) $THLNs$. Let $G_n$ be an $n$-$D$ $THLN$ and $F$ be a subset of $V(G_n)\cup E(G_n)$ with $|F|\leq n-2$. We show that for arbitrary two different correct vertices $u$ and $v$, there is a faultless path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^n-f_v-1-\alpha$, where $\alpha=0$ if vertices $u$ and $v$ form a normal vertex-pair and $\alpha=1$ if vertices $u$ and $v$ form a weak vertex-pair in $G_n-F$($n\geq5$).

Full text

1 Introduction

The $n$-dimensional hypercube[2], which possesses many outstanding properties such as recursive structure, relatively small degree, high symmetry, effective routing and broadcasting algorithms[3], is one of the most efficient, versatile interconnection network and, thus, becomes the preferred topological structure of parallel processing and parallel computing systems[5, 4]. Although hypercube networks have many excellent properties, it is well known that they also have inherent shortcomings, such as large diameter. Therefore, many scholars have proposed some hypercube variants, aiming at improving the defects of hypercubes, such as Efe’s crossed cubes[10], Cull’s and Larson’s Mobius cubes[7], Hilbers’s twisted cubes[9], Yang’s locally twisted cubes[8]. These hypercube variants retain the good properties of hypercubes, but also have many properties superior to hypercubes, such as the diameter of hypercube variants is almost half of the diameter of hypercubes.

Linear arrays (i.e. paths), rings (i.e. cycles), trees and meshes are the common data structures or foundational interconnection structures used in parallel computing. The hypercubes and hypercube variants can embed paths[34, 35, 36], cycles[37, 38], trees[40, 39], meshes[42, 41, 43]. In the process of large-scale Internet operation, it is inevitable that various errors may occur at nodes and edges. It is significant to find an embedding of a guest graph into a host graph where all faulty nodes and edges have been removed. This is called fault-tolerant embedding. Much work has been done on the fault-tolerant embedding[11, 12, 13, 14, 31, 32, 33, 15, 17, 18, 19, 20, 21, 24, 26, 27, 28, 25, 29, 30, 16]. A survey paper of Xu and Ma [12] lists many results on this topics until 2009.

The hypercube-like networks(short for $HLNs$) are a large class of network topologies [1, 13, 14]. Among $HLNs$ one may be identified as a subclass of networks, which in the paper is addressed as the twisted hypercube-like networks (short for $THLNs$), proposed by Yang[31] in 2011.

Definition 1.1**.**

[31] An $n$($n\geq 3$)-dimensional (short for $n$-$D$) twisted hypercube-like network (short for $THLN$) is a graph defined recursively as follows.

(1) A $3$-$D$ $THLN$ is isomorphic to the graph depicted in Fig.1(a).

(2) For $n\geq 4$, an $n$-$D$ $THLN$ $G_{n}$ is obtained from two vertex-disjoint $(n-1)$-$D$ $THLNs$, denoted by $G^{0}_{n-1}$ and $G^{1}_{n-1}$, in this way:

$V(G_{n})=V(G^{0}_{n-1})\cup V(G^{1}_{n-1})$,

$E(G_{n})=E(G^{0}_{n-1})\cup E(G^{1}_{n-1})\cup$

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \{(u,\phi(u)):u\in V(G^{0}_{n-1})\}$,

where $\phi:V(G^{0}_{n-1})\rightarrow V(G^{1}_{n-1})$ is a bijective mapping. In the following, we will denote this graph $G_{n}$ as $G_{n}=\oplus_{\phi}(G^{0}_{n-1},G^{1}_{n-1}$). Fig.1(b) plots a $4D$ $THLN$.**

Specifically, the fore-mentioned hypercube variant networks are all $THLNs$. In 2005, Park et al.[13] demonstrated that all $n$-$D$ $THLNs$ are Hamiltonian with at most $n-2$ faulty elements and Hamiltonian connected with at most $n-3$ faulty elements. Furthermore, Zhang et al.[15] improved the upper bound of fault tolerant Hamiltonian connectivity to $n-2$ excepting only a pair of vertices and gave the definitions of weak vertex-pair and normal vertex-pair as follows.

Definition 1.2**.**

[15]* Let $F\subset V(G_{n})\cup E(G_{n})$ with $|F|=n-2$. If $G_{n}-F$ contains a vertex $w$ such that $N_{G_{n}-F}(w)=\{w_{1},w_{2}\}$, then $w$ is called as a weak 2-degree vertex and $(w_{1},w_{2})$ is called as a $w$-weak vertex pair(short for weak vertex pair ).*

If $F=\{a,b\}$, for instance, then $w$ is a weak 2-degree vertex and $(w_{1},w_{2})$ is a weak vertex-pair in $G_{4}-F$(See Figure 2).

Unquestionably, for the weak vertex-pair $(w_{1},w_{2})$, any correct path $P_{w_{1}w_{2}}$ of length $l\geq 3$ can’t include the weak 2-degree vertex $w$. It follows there is no correct hamiltonian path joining vertices $w_{1}$ and $w_{2}$ in $G_{n}-F$[15]. However, we proved that $G_{n}-F(n\geq 5)$ contains at most one weak $2$-degree vertex $w$ and one $w$-weak vertex-pair for any $F\subset V(G_{n})\cup E(G_{n})$ with $|F|\leq n-2$ in reference[15].

Definition 1.3**.**

[15]* If $(w_{1},w_{2})$ is not a weak vertex-pair for any vertex $w\in V(G_{n}-F)$, then $(w_{1},w_{2})$ is addressed as a normal vertex pair.*

Theorem 1.1**.**

[15]Let $F\subset V(G_{n})\cup E(G_{n})$ with $|F|\leq n-2$. Then for any vertex-pair $(u,v)$ in $G_{n}-F$, there is a $(n-2)$-fault-tolerant hamiltonian path $P_{uv}$ except $(u,v)$ being a weak vertex-pair.

In the paper, we studied the path-embedding in a $THLN$ with $n-2$ faulty elements and showed that if $F\subset V(G_{n})\cup E(G_{n})$ and $|F|\leq n-2$, then for arbitrary two different correct vertices $u$ and $v$, there is a fault-free path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^{n}-f_{v}-1-\alpha$, where $\alpha=0$ if vertices $u$ and $v$ form a normal vertex-pair and $\alpha=1$ if vertices $u$ and $v$ form a weak vertex-pair in $G_{n}-F$($n\geq 5$).

To do this simply, we can denote $G_{n}=L\oplus R$, where $L=G_{n-1}^{0}$ and $R=G_{n-1}^{1}$. For any vertex $x\in L({\rm or}\ R)$, let $x^{R}({\rm or}\ x^{L})$ be the sole vertex adjacent to vertex $x$ in $R({\rm or}\ L)$, and $N_{L}(x)({\rm or}\ N_{R}(x))$ be the set of vertices that are adjacent to vertex $x$ in $L({\rm or}\ R)$. Let $E^{C}$ be the set of edges that join $L$ to $R$ and $E_{L}(x)({\rm or}\ E_{R}(x))$ be the set of edges incident to vertex $x$ in $L({\rm or}\ R)$.

We use $P_{uv}$ to represent the path from vertex $u$ to vertex $v$. If $P_{uw}=(u,u_{1},\cdots,u_{s},w)$, $P_{wv}=(w,w_{1},\cdots,w_{t},v)$ and $V(P_{uw})\cap V(P_{wv})=\{w\}$, we use $P_{uw}+P_{wv}$ to denote the path $P_{uv}=(u,u_{1},\cdots,u_{s},w,w_{1},\cdots,w_{t},v)$, $P_{uv}(u_{1},w_{1})$ to represent the subpath of $P_{uv}$ which is from vertex $u_{1}$ to vertex $w_{1}$, $l_{uv}$ to denote the length of $P_{uv}$, $d_{uv}$ to denote the distance between vertex $u$ to vertex $v$. We denote $F^{L}=F\cap L$, $F^{R}=F\cap R$, $F^{C}=F\cap E^{C}$, $F_{v}=F\cap V(G_{n})$, $F_{e}=F\cap E(G_{n})$, $f_{v}=|F_{v}|$, $f_{v}^{L}=|F_{v}\cap V(L)|$, $f_{v}^{R}=|F_{v}\cap V(R)|$. We have $f_{v}=f_{v}^{L}+f_{v}^{R}$.

This paper is organized as below. Section 2 proved the main result. Section 3 concludes the paper.

2 Main Result

In the section, we will establish the main result of the paper. We depict theorem 2.1 as follows.

Theorem 2.1**.**

If $F\subset V(G_{n})\cup E(G_{n})$ and $|F|\leq n-2$, then for any two distinct fault-free vertices $u$ and $v$, there exists a fault-free path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^{n}-f_{v}-1-\alpha$, where $\alpha=0$ if vertices $u$ and $v$ form a normal vertex-pair and $\alpha=1$ if vertices $u$ and $v$ form a weak vertex-pair in $G_{n}-F$($n\geq 5$).

Proof.

We prove the theorem by the induction on $n\geq 5$. The result holds for $n=5$ by developing computer program using depth first searching technique combining with backtracking and branch and bound algorithm. Assume that the theorem holds for $n-1$ with $n\geq 6$, then we must show the theorem holds for $n$. In general, we assume $|F^{R}|\leq|F^{L}|$. Then $|F^{R}|\leq\lfloor\frac{n-2}{2}\rfloor\leq n-4$. Since for any vertex $x\in R$, $|N_{R}(x)|=n-1$. By $|F^{R}|\leq n-4$, $|N_{R-F^{R}}(x)|\geq 3$. Then there is no weak vertex-pair in $R-F^{R}$.

Let $u,v$ be any two distinct fault-free vertices in $G_{n}-F$. By Theorem 1.1, there is a faultless path $P_{uv}$ of length $l=2^{n}-f_{v}-1$ if vertices $u$ and $v$ form a normal vertex-pair in $G_{n}-F$. Then we only need to find each length $l$ with $2^{n-1}-1\leq l\leq 2^{n}-f_{v}-2$ between arbitrary different vertices $u$ and $v$ in $G_{n}-F$. We divide the proof to two cases: (1), $|F^{L}|\leq n-3$; (2), $|F^{L}|=n-2$.

Case 1. $|F^{L}|\leq n-3$.

Case 1.1. $u,v\in V(L-F^{L})$ or $u,v\in V(R-F^{R})$. Firstly, We prove the case of $u,v\in V(L-F^{L})$.

Since $|F^{L}|\leq n-3$, by induction hypothesis, there is a faultless path $P_{uv}$ of each length $l$ with $2^{n-2}-1\leq l\leq 2^{n-1}-f_{v}^{L}-2$ in $L-F^{L}$. Notice that there exist $\lfloor\frac{l+1}{2}\rfloor$ vertex-pairs in $P_{uv}$. Since $\lfloor\frac{l+1}{2}\rfloor-(n-2)\geq\frac{2^{n-2}}{2}-(n-2)\geq 4(n\geq 6)$, there is a faultless edge $ab\in E(P_{uv})$ with $a^{R},b^{R},aa^{R},bb^{R}\notin F$. Since $|F^{R}|\leq n-4$, by induction hypothesis, there is a faultless path $P_{a^{R}b^{R}}$ of each length $l_{a^{R}b^{R}}$ with $2^{n-2}-1\leq l_{a^{R}b^{R}}\leq 2^{n-1}-f_{v}^{R}-1$ in $R-F^{R}$. Let $P^{1}_{uv}=P_{uv}(u,a)+aa^{R}+P_{a^{R}b^{R}}+b^{R}b+P_{uv}(b,v)$. Then $P^{1}_{uv}$ is a faultless path of length $l^{1}_{uv}$ with $2^{n-1}-1\leq l^{1}_{uv}\leq 2^{n}-f_{v}-2$ in $G_{n}-F$(see Fig.3(a)).

For $u,v\in V(R-F^{R})$, by a similar discussion, we can get a faultless path $P^{1}_{uv}$ of each length $l^{1}_{uv}$ with $2^{n-1}-1\leq l^{1}_{uv}\leq 2^{n}-f_{v}-2$ in $G_{n}-F$.

Case 1.2. $u\in V(L-F^{L})$ and $v\in V(R-F^{R})$.

By the definition of $G_{n}$, $|E^{C}|=2^{n-1}$. Since $2^{n-1}-(n-2)\geq 28(n\geq 6)$, there is a faultless edge $ab$ with $ab\in E^{C}$, $a,b\notin\{u,v\}$ and $a,b\notin F$. By induction hypothesis, there is a faultless path $P_{ua}$ of each length $l_{ua}$ with $2^{n-2}-1\leq l_{ua}\leq 2^{n-1}-f_{v}^{L}-2$ in $L-F^{L}$ and a faultless path $P_{bv}$ of each length $l_{bv}$ with $2^{n-2}-1\leq l_{bv}\leq 2^{n-1}-f_{v}^{R}-1$ in $R-F^{R}$. Let $P_{uv}=P_{ua}+ab+P_{bv}$. Then $P_{uv}$ is a faultless path of each length $l_{uv}$ with $2^{n-1}-1\leq l_{uv}\leq 2^{n}-f_{v}-2$ in $G_{n}-F$(see Fig.3(b)).

Case 2. $|F^{L}|=n-2$. Then $|F^{R}|=|F^{C}|=0$.

Case 2.1. $|F^{L}\cap V(L)|\geq 1$. Let $x\in F^{L}\cap V(L)$.

Case 2.1.1. $u,v\in V(L-F^{L})$.

We mark the faulty vertex $x$ as faultless temporarily. Let $F^{L}_{1}=F^{L}-x$, then $|F_{1}^{L}|=|F^{L}|-1=n-3$. By induction hypothesis, there is a faultless path $P_{uv}$ of each length $l_{uv}$ with $2^{n-2}-1\leq l_{uv}\leq 2^{n-1}-(f_{v}^{L}-1)-2=2^{n-1}-f_{v}^{L}-1$ in $L-F^{L}_{1}$. If the path $P_{uv}$ contains the faulty vertex $x$, let $a,b\in N_{P_{uv}}(x)$; otherwise, we can arbitrarily select a vertex $c$ from the path $P_{uv}$. Let $a,b\in N_{P_{uv}}(c)$. Since $|F^{R}|=0$, by induction hypothesis, there is a faultless path $P_{a^{R}b^{R}}$ of each length $l_{a^{R}b^{R}}$ with $2^{n-2}-1\leq l_{a^{R}b^{R}}\leq 2^{n-1}-1$ in $R$. Let $P^{1}_{uv}=P_{uv}(u,a)+aa^{R}+P_{a^{R}b^{R}}+b^{R}b+P_{uv}(b,v)$. Then $P^{1}_{uv}$ is a faultless path of each length $l^{1}_{uv}$ with $2^{n-1}-1\leq l^{1}_{uv}\leq 2^{n}-f_{v}-2$ in $G_{n}-F$(see Fig.4).

Case 2.1.2. $u\in V(L-F^{L})$ and $v\in V(R)$.

We mark the faulty vertex $x$ as faultless temporarily. Let $F^{L}_{1}=F^{L}-x$, then $|F^{L}_{1}|=|F^{L}|-1=n-3$. By induction hypothesis, there is a faultless path $P_{ux}$ of each length $l_{ux}$ with $2^{n-2}-1\leq l_{ux}\leq 2^{n-1}-(f_{v}^{L}-1)-2=2^{n-1}-f_{v}^{L}-1$ in $L-F^{L}_{1}$. Let $x_{1}\in N_{P_{ux}}(x)$.

Case 2.1.2.1. $x_{1}^{R}=v$.

Let $ab\in E(P_{ux})$ with $a,b\notin\{u,x_{1},x\}$. We mark the correct vertex $v$ as faulty temporarily. Let $F^{R}_{1}=F^{R}+v$, then $|F^{R}_{1}|=|F^{R}|+1\leq n-4(n\geq 6)$. By induction hypothesis, there is a faultless path $P_{a^{R}b^{R}}$ of each length $l_{a^{R}b^{R}}$ with $2^{n-2}-1\leq l_{a^{R}b^{R}}\leq 2^{n-1}-2$ in $R-F^{R}_{1}$. Let $P_{uv}=P_{ux}(u,a)+aa^{R}+P_{a^{R}b^{R}}+b^{R}b+P_{ux}(b,x_{1})+x_{1}v$. Then $P_{uv}$ is a faultless path of each length $l_{uv}$ with $2^{n-1}-1\leq l_{uv}\leq 2^{n}-f_{v}-2$ in $G_{n}-F$(see Fig.5(a)).

Case 2.1.2.2. $x_{1}^{R}\neq v$.

By induction hypothesis, there is a faultless path $P_{x_{1}^{R}v}$ of each length $l_{x_{1}^{R}v}$ with $2^{n-2}-1\leq l_{x_{1}^{R}v}\leq 2^{n-1}-1$ in $R$. Let $P_{uv}=P_{ux}(u,x_{1})+x_{1}x_{1}^{R}+P_{x_{1}^{R}v}$. Then $P_{uv}$ is a faultless path of each length $l_{uv}$ with $2^{n-1}-1\leq l_{uv}\leq 2^{n}-f_{v}-2$ in $G_{n}-F$(see Fig.5(b)).

Case 2.1.3. $u,v\in V(R)$.

Since $|F^{R}|=0$, by induction hypothesis, there is a faultless path $P_{uv}$ of length $l=2^{n-1}-1$ in $R$. Thus, we only need to consider each length $l$ with $2^{n-1}\leq l_{uv}\leq 2^{n}-f_{v}-2$.

Case 2.1.3.1. $|\{u^{L},v^{L}\}\cap F_{v}|\geq 1$. In general, assume $u^{L}\in F_{v}$. We mark the faulty vertex $u^{L}$ as faultless temporarily. Let $F^{L}_{1}=F^{L}-u^{L}$, then $|F^{L}_{1}|=|F^{L}|-1=n-3$.

Let $S=N_{R}(v)-u$. Then $|S|\geq n-2$. Since $|F^{L}_{1}|=n-3$, there is a vertex $v_{1}\in S$ with $v_{1}^{L}\notin F$. By induction hypothesis, there is a faultless path $P_{u^{L}v_{1}^{L}}$ of each length $l_{u^{L}v_{1}^{L}}$ with $2^{n-2}-1\leq l_{u^{L}v_{1}^{L}}\leq 2^{n-1}-(f_{v}^{L}-1)-2=2^{n-1}-f_{v}^{L}-1$ in $L-F_{1}^{L}$. Let $u_{1}\in N_{P_{u^{L}v_{1}^{L}}}(u^{L})$.

If $u_{1}^{R}\neq v$, let $F_{1}^{R}=F^{R}+\{v_{1},v\}$, then $|F_{1}^{R}|=|F^{R}|+2=2\leq n-4(n\geq 6)$. By induction hypothesis, there is a faultless path $P_{uu_{1}^{R}}$ of each length $l_{uu_{1}^{R}}$ with $2^{n-2}-1\leq l_{uu_{1}^{R}}\leq 2^{n-1}-3$ in $R-F_{1}^{R}$. Let $P_{uv}=P_{uu_{1}^{R}}+u_{1}^{R}u_{1}+P_{u^{L}v_{1}^{L}}(u_{1},v_{1}^{L})+v_{1}^{L}v_{1}+v_{1}v$. Then $P_{uv}$ is a faultless path of each length $l_{uv}$ with $2^{n-1}\leq l_{uv}\leq 2^{n}-f_{v}-2$ in $G_{n}-F$(See Fig.6(a)).

If $u_{1}^{R}=v$, let $F^{R}_{1}=F^{R}+v$, then $|F^{R}_{1}|=1\leq n-4(n\geq 6)$. By induction hypothesis, there is a faultless path $P_{uv_{1}}$ of each length $l_{uv_{1}}$ with $2^{n-2}-1\leq l_{uv_{1}}\leq 2^{n-1}-2$ in $R-F^{R}_{1}$. Let $P_{uv}=P_{uv_{1}}+v_{1}v^{L}_{1}+P_{u^{L}v_{1}^{L}}(v_{1}^{L},u_{1})+u_{1}v$. Then $P_{uv}$ is a fault-free path of each length $l_{uv}$ with $2^{n-1}\leq l_{uv}\leq 2^{n}-f_{v}-2$ in $G_{n}-F$(See Fig.6(b)).

Case 2.1.3.2. $|\{u^{L},v^{L}\}\cap F_{v}|=0$. We mark the faulty vertex $x$ as faultless temporarily. Let $F^{L}_{1}=F^{L}-x$, then $|F^{L}_{1}|=|F^{L}|-1=n-3$.

Since $|F^{L}_{1}|=n-3$, by induction hypothesis, there is a faultless path $P_{xv^{L}}$ of each length $2^{n-2}-1\leq l_{xv^{L}}\leq 2^{n-1}-(f_{v}^{L}-1)-2=2^{n-1}-f_{v}^{L}-1$ in $L-F_{1}^{L}$. Let $x_{1}\in N_{P_{xv^{L}}}(x)$.

If $x_{1}^{R}=u$, let $F_{1}^{R}=F^{R}+\{u,v\}$, then $|F_{1}^{R}|=|F^{R}|+2=2\leq n-4(n\geq 6)$. Let $ab\in E(P_{xv^{L}})$ with $a,b\notin\{x,x_{1},v^{L}\}$. By induction hypothesis, there is a faultless path $P_{a^{R}b^{R}}$ of each length $l_{a^{R}b^{R}}$ with $2^{n-2}-1\leq l_{a^{R}b^{R}}\leq 2^{n-1}-3$ in $R-F_{1}^{R}$. Let $P_{uv}=ux_{1}+P_{xv^{L}}(x_{1},a)+aa^{R}+P_{a^{R}b^{R}}+b^{R}b+P_{xv^{L}}(b,v^{L})+v^{L}v$. Then $P_{uv}$ is a faultless path of each length $l_{uv}$ with $2^{n-1}\leq l_{uv}\leq 2^{n}-f_{v}-2$ in $G_{n}-F$(See Fig.7(a)).

If $x_{1}^{R}\neq u$, let $F_{1}^{R}=F^{R}+v$, then $|F_{1}^{R}|=|F^{R}|+1=1\leq n-4(n\geq 6)$. By induction hypothesis, there is a faultless path $P_{ux_{1}^{R}}$ of each length $l_{ux_{1}^{R}}$ with $2^{n-2}-1\leq l_{ux_{1}^{R}}\leq 2^{n-1}-2$ in $R-F_{1}^{R}$. Let $P_{uv}=P_{ux_{1}^{R}}+x_{1}^{R}x_{1}+P_{xv^{L}}(x_{1},v^{L})+v^{L}v$. Then $P_{uv}$ is a faultless path of each length $l_{uv}$ with $2^{n-1}\leq l_{uv}\leq 2^{n}-f_{v}-2$ in $G_{n}-F$(See Fig.7(b)).

Case 2.2. $|F^{L}\cap V(L)|=0$. Then $F^{L}=F_{e}=F$.

Let $x\in V(G_{n})$ with $d_{G_{n}-F}(x)=\delta(G_{n}-F)$.

Case 2.2.1. $x\notin\{u,v\}$.

Let $e$ be an edge with $e\in F_{e}$, $F^{1}=F-e+\{x\}$, then $|F^{1}|=n-2$ and $|F^{1}_{v}|=1$. We show that $(u,v)$ is a normal vertex pair in $G_{n}-F^{1}$ as follows.

If $\delta(G_{n}-F)\geq 4$, we discuss $\delta(G_{n}-F^{1})$ in the following four cases.

(1) For any correct vertex $x_{1}\in N_{G_{n}-F}(x)$ with $e\notin E_{G_{n}}(x_{1})$. Notice that $\delta(G_{n}-F)\geq 4$, then $d_{G_{n}-F^{1}}(x_{1})=d_{G_{n}-F}(x_{1})-1\geq 3$.

(2) For any correct vertex $x_{1}\in N_{G_{n}-F}(x)$ with $e\in E_{G_{n}}(x_{1})$. Since $F^{1}=F-e+\{x\}$, we have $d_{G_{n}-F^{1}}(x_{1})=d_{G_{n}-F}(x_{1})\geq 4$.

(3) For any correct vertex $x_{1}\notin N_{G_{n}-F}(x)$ with $e\in E_{G_{n}}(x_{1})$. Notice that $\delta(G_{n}-F)\geq 4$, then $d_{G_{n}-F^{1}}(x_{1})=d_{G_{n}-F}(x_{1})+1\geq 5$.

(4) For any correct vertex $x_{1}\notin N_{G_{n}-F}(x)$ with $e\notin E_{G_{n}}(x_{1})$, Since $F^{1}=F-e+\{x\}$, we have $d_{G_{n}-F^{1}}(x_{1})=d_{G_{n}-F}(x_{1})\geq 4$.

Above all, we conclude that $\delta(G_{n}-F^{1})\geq 3$.

If $\delta(G_{n}-F)\leq 3$, then $|E_{G_{n}}(x)\cap F|\geq n-3$. For any $z\in V(G_{n}-F^{1})$, since $|F|=n-2$ and $|E_{G_{n}}(x)\cap E_{G_{n}}(z)|\leq 1$, we have $|(E_{G_{n}}(z)\cup N_{G_{n}}(z))\cap F^{1}|\leq 3$. It follows that $\delta(G_{n}-F^{1})\geq n-3\geq 3(n\geq 6)$.

Hence, there is no weak vertex-pair in $G_{n}-F^{1}$, i.e., $(u,v)$ is a normal vertex pair in $G_{n}-F^{1}$. By the proof of Case 2.1 and Theorem 1.1, there is a faultless path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^{n}-|F^{1}_{v}|-1=2^{n}-2$ in $G_{n}-F$($n\geq 5$).

Case 2.2.2. $x\in\{u,v\}$. In general, assume that $x=u$.

Let $e$ be an edge with $e=uy\in F_{e}$ with $y\neq v$ and $F^{1}=F-e+\{y\}$, then $|F^{1}|=n-2$ and $|F^{1}_{v}|=1$. We show that $(u,v)$ is a normal vertex pair in $G_{n}-F^{1}$ as follows.

Let $z$ be an arbitrary vertex of $V(G_{n}-F^{1})-\{u,v\}$.

If $\delta(G_{n}-F)\geq 4$, similar to the above discussion in Case 2.2.1, we have $\delta(G_{n}-F^{1})\geq 3$. It means that $d_{G_{n}-F^{1}}(z)\geq 3$.

If $\delta(G_{n}-F)\leq 3$, then $|E_{G_{n}}(u)\cap F|\geq n-3$. Since $|F|=n-2$ and $|E_{G_{n}}(u)\cap E_{G_{n}}(z)|\leq 1$, we have $|(E_{G_{n}}(z)\cup N_{G_{n}}(z))\cap F^{1}|\leq 3$. It follows that $d_{G_{n}-F^{1}}(z)\geq n-3\geq 3(n\geq 6)$.

Hence, $(u,v)$ can not be a $z$-weak vertex pair in $G_{n}-F^{1}$, i.e., $(u,v)$ is a normal vertex pair in $G_{n}-F^{1}$. By the proof of Case 2.1 and Theorem 1.1, there is a faultless path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^{n}-|F^{1}_{v}|-1=2^{n}-|F^{1}_{v}|-1=2^{n}-2$ in $G_{n}-F$($n\geq 5$). ∎

3 Concluding Remarks

This paper considered the path-embedding in an $n$-$D$ $THLN$($n\geq 5$) with a set $F$ of up to $n-2$ faulty elements. We have proved that for arbitrary two different correct vertices $u$ and $v$, there exists a fault-free path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^{n}-f_{v}-1-\alpha$, where $\alpha=0$ if vertices $u$ and $v$ form a normal vertex-pair and $\alpha=1$ if vertices $u$ and $v$ form a weak vertex-pair in $G_{n}-F$($n\geq 5$). The proposed theorem in the paper can be applied to several multiprocessor systems, including $n$-dimensional Möbius cubes $MQ_{n}$[7], $n$-dimensional locally twisted cubes $LTQ_{n}$[8], $n$-dimensional twisted cubes $TQ_{n}$[9] for odd $n$, and $n$-dimensional crossed cubes $CQ_{n}$[10]. Fig.8 illustrates $CQ_{3}$ and $CQ_{4}$. The graphs shown in Fig.9 are $LTQ_{3}$ and $LTQ_{4}$. Fig.10 plots $MQ^{0}_{3}$, $MQ^{0}_{4}$, $MQ^{1}_{3}$ and $MQ^{1}_{4}$. The graphs shown in Fig.11 are $TQ_{3}$ and $TQ_{5}$. By the discussion in reference[15], $MQ_{n},LTQ_{n},TQ_{n},CQ_{n}\in THLNs$.

Hence, by Theorem 2.1, we can obtain the following four Corollaries.

Corollary 3.1**.**

If $F\subset V(CQ_{n})\cup E(CQ_{n})$ with $|F|\leq n-2$, then for any two different correct vertices $u$ and $v$, there is a fault-free path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^{n}-f_{v}-1-\alpha$, where $\alpha=0$ if vertices $u$ and $v$ form a normal vertex-pair and $\alpha=1$ if vertices $u$ and $v$ form a weak vertex-pair in $CQ_{n}-F$($n\geq 5$).**

Corollary 3.2**.**

If $F\subset V(LTQ_{n})\cup E(LTQ_{n})$ with $|F|\leq n-2$, then for any two different correct vertices $u$ and $v$, there exists a fault-free path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^{n}-f_{v}-1-\alpha$, where $\alpha=0$ if vertices $u$ and $v$ form a normal vertex-pair and $\alpha=1$ if vertices $u$ and $v$ form a weak vertex-pair in $LTQ_{n}-F$($n\geq 5$). **

Corollary 3.3**.**

If $F\subset V(MQ_{n})\cup E(MQ_{n})$ with $|F|\leq n-2$, then for any two different correct vertices $u$ and $v$, there exists a fault-free path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^{n}-f_{v}-1-\alpha$, where $\alpha=0$ if vertices $u$ and $v$ form a normal vertex-pair and $\alpha=1$ if vertices $u$ and $v$ form a weak vertex-pair in $MQ_{n}-F$($n\geq 5$). **

Corollary 3.4**.**

If $F\subset V(TQ_{n})\cup E(TQ_{n})$ with $|F|\leq n-2$, then for any two different correct vertices $u$ and $v$, there exists a fault-free path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^{n}-f_{v}-1-\alpha$, where $\alpha=0$ if vertices $u$ and $v$ form a normal vertex-pair and $\alpha=1$ if vertices $u$ and $v$ form a weak vertex-pair in $TQ_{n}-F$($n\geq 5$) for any odd $n$. **

In this paper, we apply our strategy to these four network topologies($MQ_{n},LTQ_{n},\\ TQ_{n},CQ_{n}$). In the future work, we will extend our strategy to other graphs of Hypercube-Like Networks.