連結網路上的連通性相關之研究

行政院國家科學委員會專題研究計畫 期中進度報告

連結網路上的連通性相關之研究(第 1 年)

期中進度報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 96-2221-E-009-137-MY3

執 行 期 間 ： 96 年 08 月 01 日至 97 年 07 月 31 日

執 行 單 位 ： 國立交通大學資訊工程學系(所)

計 畫 主 持 人 ： 譚建民

計畫參與人員： 教授-主持人(含共同主持人)：譚建民

處 理 方 式 ： 期中報告不提供公開查詢

中 華 民 國 97 年 05 月 27 日

1

行政院國家科學委員會專題研究計畫 期中進度報告

計畫類別： 一般型研究計畫（個別型）

計畫編號： NSC 96-2221-E-009-137-MY3

執行期間： 96年 8月 1日 至 99年 7月31日

執行單位： 國立交通大學資訊工程學系(所)

計畫主持人： 譚建民

計畫參與人員：施倫閔、江玠峰、蔡宗翰、石圜鋼、林政寬、龔自良。

報告類型：精簡報告

報告附件：發表論文

處理方式：本計畫可公開查詢

中華民國九十七年五月二十七日

連結網路上的連通性相關之研究

行政院國家科學委員會補助專題研究計畫期中報告

※※※※※※※※※※※※※※※※※※※※※※※※

※

※

※

_{連 結 網 路 上 連 通 性 之 研 究}

※

※

※

※※※※※※※※※※※※※※※※※※※※※※※※

計畫類別：■個別型計畫 □整合型計畫

計畫編號：96-2221-E-009-137-MY3

執行期間：96年 8月 1日至 99年 7月 31日

計畫成果：本計畫結果由計畫主持人及博士班研究生共同發表，已刊

登在2008年Information Processing Letters期刊，發表的論文

為：

Lun-Min Shih, Chieh-Feng Chiang, Lih-Hsing Hsu, and

Jimmy J.M. Tan “Strong Menger Connectivity with

Conditional Faults on the Class of Hypercube-like Networks”,

Information Processing Letters, (2008), 106, pp. 64-69.

計畫主持人：譚建民

成果報告類型(依經費核定清單規定繳交)：■精簡報告 □完整報告

本成果報告包括以下應繳交之附件：

□赴國外出差或研習心得報告一份

□赴大陸地區出差或研習心得報告一份

□出席國際學術會議心得報告及發表之論文各一份

□國際合作研究計畫國外研究報告書一份

執行單位：國立交通大學資訊工程學系

中華民國九十七年五月二十七日

3

連結網路上的連通性相關之研究

A Study on the Connected Property of Interconnection Networks

計劃編號：96-2221-E-009-137-MY3

計劃期限：96/8/1～99/7/31

主持人：譚建民 國立交通大學資訊工程學系 教授

一、

中英文摘要

中文摘要

連結網路一個相當廣泛的研究領域。對連

結網路的結構時常以圖形G來表示。連通度

是在連結網路的研究領域中重要的研究主

題之一；對一個圖形G的連通度我們以κ(G)

來表示，其定義為一個連通的圖形中移除

最少的節點數後會變為不連通。在古典的

定理中，Menger提出局部性的連通度的觀

念；在此，我們在特定的N維超立方體家族

的網路中，對局部性的連通度也做了更進

一步的研究。

N 維 度 的 超 立 方 體 家 族 的 網 路 其 建 構 方

式，是將兩個N-1維度的超立方體家族的網

路以一種完全配對的方式將兩個網路連結

起來。在此計劃中，我們研究在N維的超立

方體家族網路中，移除N-2個節點之後，對

任 何 兩 個 節 點 u,v 之 間 都 有

min{deg(u),deg(v)}條節點獨立的路徑；

其中deg(u)及deg(v)是代表N維超立方體

家族網路中移除部分節點數後，u和v各別

的分支度數目。這主題在OH及Chen[6]的文

章中有相關的研究。我們提出了另外的方

法，更簡單地去証明這個特性。不僅如此，

如果我們將移除的節點加一些條件，規定

移除後的圖形每個節點都至少要有兩個好

的節點相鄰。在這個條件之下，所允許可

以移除的節點數個數將會提昇到2N-5，而

且在移除這些節點之後，圖形仍然具有局

部連通度的性質。對於超立方體家族網路

而言，移除2N-5的節點數是具有最佳情形

的容錯性質了。

關鍵詞：局部連通度、條件式容錯、超立

方體家族、連結網路。

英文摘要

Interconnection networks have been

widely studied recently. The architecture of

an interconnection network is usually

denoted as an undirected graph G. Among all

fundamental properties for interconnection

networks, the (vertex) connectivity is a major

parameter widely discussed for the

connection status of networks. A basic

definition of the connectivity κ (G) of a

graph G is defined as the minimum number

of vertices whose removal from G produces

a disconnected graph. In contrast to this

concept, Menger [5] provided a local point of

view, and define the connectivity of any two

vertices as the minimum number of

internally vertex-disjoint paths between them.

In this project, we study the Menger property

on a class of hypercube-like networks [9],

which is a variation of the classical

hypercube network by twisting some pairs of

links in it. We show that in all n-dimensional

hypercube-like networks with some vertices

removed, every pair of unremoved vertices u

and v are connected by min{deg(u); deg(v)}

vertex-disjoint paths, where deg(u) and deg(v)

are the remaining degree of vertices u and v,

respectively. This concept is firstly applied

on hypercubes and stars by Oh and Chen [6,

7, 8]. In this project, we give a simpler proof

of this result. Furthermore, if we restrict a

condition such that each vertex has at least

two fault-free adjacent vertices, all

hypercube-like networks still have this

strong Menger property, even if there are up

to 2n - 5 vertex faults. The bound of 2n - 5 is

sharp.

Keywords: strong Menger connectivity,

conditional faults, hypercube-like network

二、

計劃緣由及目的

The topology of a multiprocessor system can

be modeled as an undirected graph

G

=

(V,E)

,

where

V(G)

represents the set of all

processors and

E(G)

represents the set of all

connecting links between the processors. For

a subset of vertices

F

⊂

V(G)

, the induced

graph obtained by deleting the vertices of

F

from

G

is denoted by

G

−

F

. Let

u

be a

vertex, we use

N(u)

to denote the set of

vertices adjacent to

u

, and use deg

(u)

to

denote the cardinality of

N(u)

. For a set of

vertices

V

’, the neighborhood of

V

’ is defined

as the set

N(V

’

)

= {U

N(v)

} −

V

’. Let

G

be a graph with a set

F

of faulty vertices, the

number of fault-free neighbors of

u

in

G

−

F

is denoted by deg

G

−

F (u)

.

Let

G

=

(V

,E

)

and

G

=

(V

,E

)

be two

disjoint graphs with the same number of

vertices. A one-to one connection between

V(G

)

_and

V(G

)

_{is defined as an edge set}

M

=

{

(v, φ (v))

|

v

∈

V

, φ (v)

∈

V

_and

φ

_:

V

→

V

_{is a bijection}. We use}

G

⊕

G

to denote the graph

G

=

(V

∪

V

,E

∪

E

∪

M)

. Different bijection functions

φ

lead to

different operations ⊕

and generate

different graphs.

The hypercube network is one of the popular

topologies in interconnection networks.

Several variants of hypercubes are proposed

by twisting some pairs of links in hypercubes,

including twisted cubes [1,4], Möbius cubes

[2], and crossed cubes [3], to name a few. To

make a unified study on these variants,

Vaidya et al. [9] proposed a class of graphs,

called a class of hypercube-like networks.

We now give a recursive definition of the

n

dimensional hypercube-like networks HL

n

as follows:

(1) HL0 =

K

1, where

K

1 is a trivial graph in

the sense that it has only one vertex; and (2)

G

∈ HL

if and only if

G

=

G

⊕

G

for

some

G

,G

∈ HL

. By the definitions

above if

G

is a graph in HL

, then

G

is a

composition of

G

⊕

G

with both

G

and

G

1

in HL

,

n≧

1. Each vertex in

G

has exactly

one neighbor in

G

.

A graph

G

is

r

-regular if the degree of every

vertex in

G

is

r

. We say that a graph

G

is

connected if there is a path between every

pair of two distinct vertices. A subset

S

of

V(G)

is a cut set if

G

−

S

is disconnected.

The connectivity of

G

, written as

κ(G)

, is

defined as the minimum size of a vertex cut

if

G

is not a complete graph, and

κ(G)

=

|

V(G)

| − 1 if otherwise. We say that a

graph

G

is

k

-connected if

k ≦ κ (G)

. In

addition, a graph has connectivity

k

if it is

k

_{-connected but not}

(k

+1

)

_-connected.

A classical theorem about connectivity was

provided by Menger as follows.

5

distinct vertices of a graph

G

and

(x, y) not

belong to

E(G)

. The minimum size of an

x,y

-cut equals the maximum number of

pairwise internally disjoint

x,y

-paths.

Following this theorem, Oh and Chen [7]

gave a definition to extend the Menger’s

theorem.

Definition 1. (See [7].) A

k

-regular graph

G

is strongly Menger-connected if for any

subgraph

G

−

F

of

G

with at most

k

− 2

vertices removed, each pair of vertices

u

and

v

in

G

−

F

are connected by min{deg

(u),

deg

(v)

} vertex-disjoint fault-free paths in

G

−

F

, where deg

(u)

and deg

(v)

are the

degree of

u

and

v

in

G

−

F

, respectively.

By Definition 1, Oh and Chen [6–8] showed

that an

n

-dimensional star graph

S

(respectively, an

n

-dimensional hypercube

Q

)

with at most

n

− 3 (respectively,

n

− 2)

vertices removed is strongly

Menger-connected. In order to be consistent

with Definition 1, we say that a graph

G

possess the strongly Menger-connected

property with respect to a vertex set

F

if,

after deleting

F

from

G

, there are min{deg

(u),

deg

(v)

} vertex-disjoint fault-free

paths connecting

u

and

v

, for each pair of

vertices

u

and

v

in

G

−

F

. Throughout this

project, we shall call a graph “strongly

Menger-connected”, and omit the description

of the remaining structure

G

−

F

of the graph,

if there is no ambiguous. It is known that the

connectivity of an

n

-dimensional

hypercube-like network HL

n

is

n

[9]. To

extend the connectivity result of HL

n

further,

we study the strongly Menger-connected

property of HL

n

with at most

n

− 2 vertices

deleted. Moreover, if we restrict a condition

such that each vertex has at least two

fault-free adjacent vertices, HL

n

still have

the strong Menger property, even if there are

up to 2

n

− 5 vertex faults.

三、

研究方法與成果

這幾年來的研究，我們在連結網路的

領域上，對一些著名的網路架構做了深入

的探討及研究。連結網路中的一些觀念如

connectivity( 連 通 度 ) 、 conditional

connectivity( 條 件 式 連 通 度 ) 、 local

connectivity（局部性連連度）我們都有

相關的研究，也將我們的成果投稿，並且

持續努力撰寫論文。在實驗室的規範中，

每週都定時研討會礒，內容如下：

一、 搜尋及集合文獻

每位學生主動從圖書館、國內外研討會及

網際網路等所收集相關主題及所需要的相

關文獻。

二、 探討文獻及發現問題

將所收集到的文獻由計畫中的成員將文獻

做進一步的分析研究，並且在每週輪流定

期報告其文獻的內容、共同分析文獻中的

主題內容，再找尋從文獻中發現可以做進

一步研究之問題，由主持人帶領成員選定

研究之主題。

三、 由指導老師帶領博士班，並由博士班

帶領碩士班共同研究，一起解決其問題，

在過程中，會需要撰寫程式來輔助定理的

證明正確性。我們也有自行發展出一些軟

體程式以供測試例子。

四、 成果發表

近年來，我們已有多篇論文被國際知

名期刊刊登。而這個計畫其中之一結果已

刊 登 在 2008 年 Information Processing

Letters期刊，發表的論文為：

Lun-Min Shih, Chieh-Feng Chiang,

Lih-Hsing Hsu, and Jimmy J.M. Tan “Strong

Menger Connectivity with Conditional Faults

on the Class of Hypercube-like Networks”,

Information Processing Letters, (2008), 106,

pp. 64-69.

四、

結論與討論

在主持人的帶領下，本計劃每週都會

討論所收集的資料，並分析和比較各種方

法的可行性。對一些有名連結網路(如：

hypercube 、 matching composition

networks 、 hypercube-like networks 、

star graph)的連通度相關問題有了更清

楚的了解，也因此為本次計劃的執行有更

良好的基礎，也順利地完成我們所預定的

研究進度。對於局部性連通度及其容錯性

質，希望可以在學術上找到新的特性，並

且可以比較分析各種不周的連結網路，去

探討它們之間的相關連通特性。期望這些

相關的問題可以提昇這方面的領域能力，

對於日後的研究有更深入的探討。

五、

參考文獻

[1] S. Abraham, K. Padmanabhan, The

twisted cube topology for multiprocessors: a

study in network asymmetry, Journal of

Parallel and Distributed Computing 13 (1991)

104–110.

[2] P. Cull, S.M. Larson, The Möbius cubes,

IEEE Transactions on Computers 44 (1995)

647–659.

[3] K. Efe, The crossed cube architecture for

parallel computing, IEEE Transactions on

Parallel and Distributed Systems 3 (1992)

513–524.

[4] A.H. Esfahanian, L.M. Ni, B.E. Sagan,

The twisted

n

-cube with application to

multiprocessing, IEEE Transactions on

Computers 40 (1991) 88–93.

[5] K. Menger, Zur allgemeinen

kurventheorie, Fund. Math. 10 (1927)

95–115.

[6] E. Oh, On strong fault tolerance (or

strong Menger-connectivity) of

multicomputer networks, PhD thesis,

Computer Science, Texas A&M University,

August 2004.

http://txspace.tamu.edu/

bitstream/1969.1/1284/1/etd-tamu-2004B-CP

SC-Oh-2.pdf.

[7] E. Oh, J. Chen, On strong

Menger-connectivity of star graphs, Discrete

Applied Mathematics 129 (2003) 499–511.

[8] E. Oh, J. Chen, Strong fault-tolerance:

Parallel routing in star networks with faults,

Journal of Interconnection Networks 4 (2003)

113–126.

[9] A.S. Vaidya, P.S.N. Rao, S.R. Shankar,

A class of hypercube-like networks, in: Proc.

of the 5th Symp. IEEE Transactions on

Parallel and Distributed Processing, Soc.,

Los Alamitos, CA, 1993, pp. 800–803.

Information Processing Letters 106 (2008) 64–69

www.elsevier.com/locate/ipl

Strong Menger connectivity with conditional faults on the class

of hypercube-like networks

Lun-Min Shih

_{, Chieh-Feng Chiang}

_{, Lih-Hsing Hsu}

_{, Jimmy J.M. Tan}

a_{Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 30050, ROC} b_{Department of Computer Science and Information Engineering, Providence University, Taichung, Taiwan 43301, ROC}

Received 18 April 2007; received in revised form 15 October 2007; accepted 16 October 2007 Available online 26 October 2007

Communicated by L. Boasson

Abstract

In this paper, we study the Menger property on a class of like networks. We show that in all n-dimensional hypercube-like networks with n− 2 vertices removed, every pair of unremoved vertices u and v are connected by min{deg(u), deg(v)} vertex-disjoint paths, where deg(u) and deg(v) are the remaining degree of vertices u and v, respectively. Furthermore, under the restricted condition that each vertex has at least two fault-free adjacent vertices, all hypercube-like networks still have the strong Menger property, even if there are up to 2n− 5 vertex faults.

©2007 Elsevier B.V. All rights reserved.

Keywords: Strong Menger connectivity; Conditional faults; Hypercube-like network; Interconnection networks

1. Introduction

Interconnection networks have been widely studied recently. The architecture of an interconnection network is usually denoted as an undirected graph G. Among all fundamental properties for interconnection networks, the (vertex) connectivity is a major parameter widely discussed for the connection status of networks. A ba-sic definition of the connectivity κ(G) of a graph G is defined as the minimum number of vertices whose removal from G produces a disconnected graph. In

con-✩ _{This work was supported in part by the National Science Council}

of the Republic of China under Contract NSC 96-2221-E-009-137-MY3.

* _{Corresponding author at: Department of Computer Science,}

Na-tional Chiao Tung University, Hsinchu City, Taiwan 30050, ROC.

E-mail address: jmtan@cs.nctu.edu.tw (J.J.M. Tan).

trast to this concept, Menger [5] provided a local point of view, and define the connectivity of any two vertices as the minimum number of internally vertex-disjoint paths between them.

In this paper, we study the Menger property on a class of hypercube-like networks [9], which is a vari-ation of the classical hypercube network by twisting some pairs of links in it. We show that in all n-di-mensional hypercube-like networks with some vertices removed, every pair of unremoved vertices u and v are connected by min{deg(u), deg(v)} vertex-disjoint paths, where deg(u) and deg(v) are the remaining degree of vertices u and v, respectively. This concept is firstly ap-plied on hypercubes and stars by Oh and Chen [6–8]. In this paper, we give a simpler proof of this result. Furthermore, if we restrict a condition such that each vertex has at least two fault-free adjacent vertices, all hypercube-like networks still have this strong Menger

0020-0190/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2007.10.009

L.-M. Shih et al. / Information Processing Letters 106 (2008) 64–69 65

property, even if there are up to 2n− 5 vertex faults. The bound of 2n− 5 is sharp.

2. Preliminary

The topology of a multiprocessor system can be modeled as an undirected graph G= (V, E), where V (G) represents the set of all processors and E(G) represents the set of all connecting links between the processors. For a subset of vertices F⊂ V (G), the in-duced graph obtained by deleting the vertices of F from G is denoted by G− F . Let u be a vertex, we use N (u)to denote the set of vertices adjacent to u, and use deg(u) to denote the cardinality of N (u). For a set of vertices V, the neighborhood of Vis defined as the set N (V)= {_v∈VN (v)} − V. Let G be a graph with a

set F of faulty vertices, the number of fault-free neigh-bors of u in G− F is denoted by degG_−F(u).

Let G0= (V0, E0)and G1= (V1, E1) be two

dis-joint graphs with the same number of vertices. A one-to-one connection between V (G0)and V (G1)is defined

as an edge set M= {(v, φ(v)) | v ∈ V0, φ(v)∈ V1and

φ: V0→ V1is a bijection}. We use G0⊕M G1 to

de-note the graph G= (V0∪ V1, E0∪ E1∪ M). Different

bijection functions φ lead to different operations ⊕M

and generate different graphs.

The hypercube network is one of the popular topolo-gies in interconnection networks. Several variants of hy-percubes are proposed by twisting some pairs of links in hypercubes, including twisted cubes [1,4], Möbius cubes [2], and crossed cubes [3], to name a few. To make a unified study on these variants, Vaidya et al. [9] pro-posed a class of graphs, called a class of hypercube-like networks. We now give a recursive definition of the n-dimensional hypercube-like networks HLn as follows:

(1) HL0= K1, where K1is a trivial graph in the sense

that it has only one vertex; and (2) G∈ HLnif and only

if G= G0⊕M G1 for some G0, G1∈ HLn₋₁. By the

definitions above if G is a graph in HLn, then G is

a composition of G0⊕MG1 with both G0 and G1 in

HLn−1, n 1. Each vertex in G0has exactly one

neigh-bor in G1.

A graph G is r-regular if the degree of every vertex in G is r. We say that a graph G is connected if there is a path between every pair of two distinct vertices. A sub-set S of V (G) is a cut sub-set if G− S is disconnected. The connectivity of G, written as κ(G), is defined as the minimum size of a vertex cut if G is not a com-plete graph, and κ(G)= |V (G)| − 1 if otherwise. We  κ(G). In

addi-A classical theorem about connectivity was provided by Menger as follows.

Theorem 1. (See [5].) Let x and y be two distinct

ver-tices of a graph G and (x, y) /∈ E(G). The minimum size of an x, y-cut equals the maximum number of pair-wise internally disjoint x, y-paths.

Following this theorem, Oh and Chen [7] gave a de-finition to extend the Menger’s theorem.

Definition 1. (See [7].) A k-regular graph G is strongly

Menger-connected if for any subgraph G− F of G with at most k− 2 vertices removed, each pair of vertices uand v in G− F are connected by min{degG−F(u),

degG_−F(v)} vertex-disjoint fault-free paths in G − F ,

where degG_−F(u) and degG_−F(v)are the degree of u

and v in G− F , respectively.

By Definition 1, Oh and Chen [6–8] showed that an n-dimensional star graph Sn (respectively, an

n-dimen-sional hypercube Qn) with at most n− 3 (respectively,

n− 2) vertices removed is strongly Menger-connected. In order to be consistent with Definition 1, we say that a graph G possess the strongly Menger-connected prop-erty with respect to a vertex set F if, after deleting F from G, there are min{degG−F(u),degG−F(v)}

vertex-disjoint fault-free paths connecting u and v, for each pair of vertices u and v in G−F . Throughout this paper, we shall call a graph “strongly Menger-connected”, and omit the description of the remaining structure G− F of the graph, if there is no ambiguous.

It is known that the connectivity of an n-dimensional hypercube-like network HLnis n [9]. To extend the

con-nectivity result of HLn further, we study the strongly

Menger-connected property of HLnwith at most n− 2

vertices deleted. Moreover, if we restrict a condition such that each vertex has at least two fault-free adja-cent vertices, HLnstill have the strong Menger property,

even if there are up to 2n− 5 vertex faults.

3. Strong Menger connectivity

In this section, we will prove that all graphs in the class of n-dimensional hypercube-like networks are strongly Menger-connected if there are at most n− 2 vertex faults. Before proving this main result, we need the following lemma, essentially it says that every

n-di-66 L.-M. Shih et al. / Information Processing Letters 106 (2008) 64–69

Lemma 1. Let G∈ HLn be an n-dimensional

hyper-cube-like network, and S be a set of vertices with|S|  2n− 3, for n  2. There exists a connected component C in G− S such that |V (C)|  2n− |S| − 1.

Proof. We prove this statement by induction on n. For

n= 2, HL2 is a cycle of length four, the result is

triv-ially true. Assume this lemma holds for n− 1, for some n 3, we will prove that it is true for n.

Let G be an n-dimensional hypercube-like network, G= G0⊕MG1, and G0, G1∈ HLn−1. Let S be a set

of vertices with|S|  2n − 3, for n  3, and let S0and

S1be subsets of set S in G0and G1, respectively. Then

|S0| + |S1| = |S|  2n − 3. Without loss of generality,

we assume |S0|  |S1|. The proof is divided into two

major cases:

Case 1: 0 |S0|  1.

Since G0is (n−1)-connected, G0−S0is connected,

for n 3. All the vertices in G0− S0 are connected

and form a connected component C0 with|V (C0)| =

2n−1− S0. By definition, all the vertices in G1− S1are

adjacent to the vertices in G0= C0∪ S0. Thus, G− S

contains a connected component C such that the number of vertices in C is greater than|V (G0)−S0|+|V (G1)−

S1| − |S0| = |V (G)| − |S| − |S0|  2n− |S| − 1. (See

Fig. 1.)

Case 2:|S0|  2 and consequently |S1|  2n − 5.

Since 2 |S0|  |S1|  2n − 5, so |S0|  n − 2 and

n 4. By induction hypothesis, there exists a connected component C1in G1−S1, and|V (C1)|  2n−1−|S1|−

1. Since the connectivity of G0is n−1 and |S0|  n−2,

G0− S0is connected. Then G− S contains a connected

component C such that the number of vertices in C is greater than |V (G0)− S0| + (|V (G1)− S1| − 1) =

|V (G)| − |S| − 1 = 2n_{− |S| − 1. 2}

By Lemma 1, we have the following corollary.

Corollary 1. Let G be an n-dimensional hypercube-like

network, n 2, and let Vbe a set of vertices in G with |V_{| = 2. Then |N(V}_{)|  2n − 2.}

Fig. 1. The illustration of the proof of Case 1 in Lemma 1.

In the following, we show that with up to n− 2 ver-tex faults, an n-dimensional hypercube-like network has strongly Menger-connected property. Referring to the relative study proposed by Oh [6], the strong Menger connectivity of regular hypercube networks has been proved. Here we provide a significantly simpler proof for the general hypercube-like networks.

Theorem 2. Consider an n-dimensional

hypercube-like network G∈ HLn, for n  2. Let F be a set

of faulty vertices with |F |  n − 2. Then each pair of vertices u and v in G − F are connected by min{deg_G−F(u),deg_G−F(v)} vertex-disjoint fault-free paths, where deg_G−F(u) and deg_G−F(v) are the re-maining degree of u and v in G− F , respectively.

Proof. Let G be an n-dimensional hypercube-like

network, and u and v be two fault-free vertices in G − F . We first assume, without loss of general-ity, that degG−F(u) degG−F(v), so min{degG−F(u),

degG−F(v)} = degG−F(u). We now show that u is

con-nected to v if the number of vertices deleted is smaller than degG−F(u)− 1 in G − F . By Theorem 1, this

implies that each pair of vertices u and v in G− F are connected by degG−F(u) vertex-disjoint fault-free

paths, where|F |  n − 2.

For the sake of contradiction, suppose that u and v are separated by deleting a set of vertices Vf, where

|Vf|  degG−F(u)− 1. As a consequence, |Vf|  n − 1

because of deg_G−F(u) deg(u)  n. Then, the sum-mation of the cardinality of these two sets F and Vf is

|F | + |Vf|  2n − 3. Let S = F ∪ Vf. By Lemma 1,

there exists a connected component C in G− S such that|V (C)|  2n−|S|−1. It means that (i) either G−S is connected, or (ii) G− S has two components, one of which contains only one vertex. If G− S is connected, it contradicts to the assumption that u and v are discon-nected. Otherwise, if G− S has two component and one of which contains only one vertex x. Since we assume that u and v are separated, one of u and v is the vertex x, say u= x. Thus, the set Vf must be the neighborhood of

uand|Vf| = degG−F(u), which is also a contradiction.

Then, u is connected to v when the number of vertices deleted is smaller than deg_G−F(u)− 1 in G − F .

The proof is complete. 2

4. Strong Menger connectivity with conditional faults

As proved in the last section, an n-dimensional hypercube-like network with at most n− 2 faulty ver-tices is strongly Menger-connected. But the result

can-L.-M. Shih et al. / Information Processing Letters 106 (2008) 64–69 67

not be guaranteed, if there are n− 1 faulty vertices and all these faulty vertices are adjacent to the same vertex. In most circumstances, the possibility of all the neigh-bors of a vertex being faulty simultaneously is very small. Motivated by the deficiency of traditional fault tolerance, we consider a measure of conditional faults by restricting that every vertex has at least two fault-free neighboring vertices.

Under this condition, we claim that for every n-di-mensional hypercube-like network with at most 2n− 5 faulty vertices and n 5, the resulting network is still strongly Menger-connected. We have an example to show that this result does not hold for n= 4. Consider a 4-dimensional HL4, this network may not be strongly

Menger-connected, if the number of conditional faults is 3. (See Fig. 2. The remaining degrees of nodes u and vare both four, with three vertices deleted as indicated in the graph. But the number of vertex-disjoint paths be-tween u and v is three.) So we can only expect the result holds for n 5.

To prove this result, we need some preliminary lemma. In the following, we show that an n-dimensional hypercube-like network with at most 3n− 6 vertex faults S has a connected component having at least 2n_{− |S| − 2 vertices.}

The proof is by induction, and the case for n= 5 is proved in the following two lemmas.

Lemma 2. Let V be a set of vertices in a 4-dimen-sional hypercube-like network with |V| = 3. Then, |N(V_{)|  7.}

Proof. Let G be a 4-dimensional hypercube-like

net-work. G is a composition of two 3-dimensional hyper-cube-like networks G0 and G1, G= G0⊕M G1, for

a matching operation ⊕M. Without loss of

general-ity, let V be a subset of V (G) containing three ver-tices {x, y, z}. If x, y, z are all in G0, by Lemma 1,

{x, y, z} has at least 4 neighboring vertices in G0.

Be-sides,{x, y, z} has 3 neighboring vertices in G1. Then,

|N({x, y, z})|  4 + 3 = 7. If x, y are in G0, and z is in

G1, by Lemma 1,{x, y} has at least 4 neighboring

ver-tices in G0. In addition,{z} has 3 neighboring vertices

in G1. Then,|N({x, y, z})|  4 + 3 = 7. 2

Lemma 3. Let G be a 5-dimensional hypercube-like

network and S be a set of vertices with|S|  9. (3n − 6= 9, for n = 5.) There exists a connected component C in G− S such that |V (C)|  25− |S| − 2.

Proof. Let G be a 5-dimensional hypercube-like

net-work, G0, G1∈ HL4, and G= G0⊕MG1, for a

match-ing operation⊕M. Let S be a set of vertices with|S| 

3n− 6 = 9, for n = 5, and let S0and S1be subsets of S

in G0and G1, respectively. Without loss of generality,

we assume|S0|  |S1|. (Note that |S|  9, so |S0|  4.)

We then consider three cases: Case 1: 0 |S0|  2.

Since G0 is (n− 1)-connected, G0− S0 is

con-nected, for n= 4. So G0− S0has only one connected

component C0with|V (C0)| = 24− S0. By definitions,

all vertices in G1− S1 are adjacent to the vertices of

G0= C ∪ S0. Let C be the connected component of

G− S containing C0. Then the number of vertices in

Cis greater than|V (G0)− S0| + |V (G1)− S1| − |S0| =

|V (G)| − |S| − |S0|  25− |S| − 2.

Case 2:|S0| = 3 and therefore |S1|  6.

G0− S0is connected by the fact that G0is (n−

1)-connected, for n 4. Thus, G0− S0has only one

con-nected component C0 with|V (C0)| = 24− S0. Then,

all vertices in G1 are connected to component C0,

ex-cept for the three vertices in G1 adjacent to the

ver-tices in S0. Since |S1|  6 and by Lemma 2, at least

one of these three vertices is connected to component G1− S1. So at least 24− |S1| − 2 vertices are connected

to component C0. Let C be the connected component

of G− S containing C0. Then, the number of vertices

in C is|V (C)|  |V (G0)− S0| + |V (G1)− S1− 2| =

|V (G)| − |S| − 2 = 25_{− |S| − 2.}

Case 3:|S0| = 4 and consequently 4  |S1|  5.

Since 5 2n − 3, for n  4. By Lemma 1, there ex-ists a connected components C0 (respectively, C1) in

G0− S0 (respectively, G1− S1) such that|V (C0)| 

24− |S0| − 1 (respectively, |V (C1)|  24− |S1| − 1).

Thus, there exists a connected component C in G− S such that|V (C)|  |V (G0)− S0− 1| + |V (G1)− S1−

1| = |V (G)| − |S| − 2 = 25_{− |S| − 2. 2}

Based on Lemma 3, the general case for n 5 is stated as follows.

68 L.-M. Shih et al. / Information Processing Letters 106 (2008) 64–69

n 5. There exists a connected component C in G − S such that|V (C)|  2n− |S| − 2.

Proof. We prove this statement by induction on n. By

Lemma 3, the result holds for n= 5. Assume the lemma holds for n− 1, for some n  6. We now show that it is true for n.

Let G be an n-dimensional hypercube-like network, G0, G1∈ HLn₋₁, and G= G0⊕MG1, for some

match-ing operation⊕M. Let S be a set of vertices with|S| 

3n− 6, for n  6, and let S0 and S1 be subsets of S

in G0 and G1, respectively. Therefore, |S0| + |S1| =

|S|  3n − 6. Without loss of generality, we assume |S0|  |S1|. The proof is divided into two major cases:

Case 1: 0 |S0|  2.

Since G0 is (n− 1)-connected, G0− S0 is

con-nected, for n 6. Let C0= G0− S0, C0is a connected

component with |V (C0)|  2n−1− S0. By definitions,

all vertices in G1− S1 are adjacent to the vertices in

G0= C0∪ S0. Let C be the connected component of

G− S containing C0. The number of vertices in C is

greater than |V (G0)− S0| + |V (G1)− S1| − |S0| =

|V (G)| − |S| − |S0|  2n− |S| − 2.

Case 2:|S0|  3 and consequently |S1|  3n − 9.

By induction hypothesis, there are two connected components C0 and C1 in G0− S0 and G1− S1, and

|V (C0)|  2n−1−|S0|−2 and |V (C1)|  2n−1−|S1|−

2, respectively. Without loss of generality, we assume that |V (C0)|  |V (C1)|. Now we focus on the

num-ber of vertices in the component C1, and discuss two

situations. First, suppose |V (C1)| = 2n−1− |S1| − 2.

By Corollary 1, |S1|  2(n − 1) − 2 = 2n − 4. So

|S0| = |S|−|S1|  n−2. Since G0is (n−1)-connected,

G0− S0is connected. G0− S0has only one connected

component C0and|V (C0)| = 2n−1− |S0|. Let C be the

connected component containing C0. Then |V (C)| =

|V (C0)| + |V (C1)|  2n−1− |S0| + 2n−1− |S1| − 2 

2n− |S| − 2. Second, suppose that |V (C1)|  2n−1−

|S1| − 1. Since |V (C0)|  |V (C1)|  2n−1− |S1| − 1,

there exists a connected component C containing C0

such that|V (C)| = |V (C0)| + |V (C1)|  2n−1− |S0| −

1+ 2n−1_{− |S}

1| − 1  2n− |S| − 2. 2

Corollary 2. Let G be an n-dimensional hypercube-like

network, n 5, and let Vbe a set of vertices in G with |V_{| = 3. Then |N(V}_{)|  3n − 5.}

As stated in the last section, we showed that every n-dimensional hypercube-like network with at most n− 2 vertex faults is strongly Menger-connected. In the following, we will show another main result that, by restricting every vertex having at least two fault-free

neighboring vertices, every n-dimensional hypercube-like network with up to 2n− 5 vertex faults is still strongly Menger-connected.

For the next theorem, we define a set of vertices Fc

in graph G to be a conditional faulty vertex set if, in the induced subgraph G− Fc, every vertex has at least two

fault-free neighboring vertices. We also call the sub-graph G− Fca conditional faulty graph.

Theorem 3. Consider an n-dimensional hypercube-like

network G∈ HLn, for n 5. Let Fc be a set of

con-ditional faulty vertices with |Fc|  2n − 5. Then each

pair of vertices u and v in G− Fc are connected by

min{degG−Fc(u),degG−Fc(v)} vertex-disjoint fault-free paths, where degG−Fc(u) and degG−Fc(v) are the de-gree of u and v in G− Fc, respectively.

Proof. Without loss of generality, we assume deg_G−Fc(u) deg_G−F

c(v), and therefore mindegG−Fc(u),degG−Fc(v)



= degG−Fc(u).

We want to prove that each pair of vertices u and v in G− Fc are connected by degG−Fc(u) vertex-disjoint fault-free paths, for |Fc|  2n − 5. We are going to

show that u is connected to v if the number of vertices deleted is smaller than degG−Fc(u)−1 in G−Fc, where |Fc|  2n − 5.

Suppose on the contrary that u and v are sepa-rated by deleting a set of vertices Vfc, where |Vfc|  degG−Fc(u) − 1. By degG−Fc(u)  deg(u)  n, we have|Vfc|  n − 1. We sum up the cardinality of these two sets Fc and Vfc. Since|Fc|  2n − 5 and |Vfc|  n− 1, then |Fc| + |Vfc|  3n − 6. Let S = Fc∪ Vfc. By Lemma 4, there exits a connected component C in G−S such that |V (C)|  2n−|S|−2 and |S|  3n−6. It means that there are at most two vertices in G− S not belonging to C. We then consider three cases:

Case 1:|V (C)| = 2n− |S|. It means that all vertices in G− S are connected, which contradicts to the as-sumption that u and v are disconnected.

Case 2:|V (C)| = 2n− |S| − 1. Only one vertex is disconnected to G−S. Since |Vfc|  degG−Fc(u)−1  degG−Fc(v)− 1, neither u nor v can be the only one disconnected vertex, a contradiction.

Case 3:|V (C)| = 2n−|S|−2. Let a and b be the two vertices in G− S not belonging to C. We consider two situations. (i) Suppose first that u∈ C. If v ∈ C, then u and v are connected, a contradiction. If v∈ {a, b}, since|Vfc|  degG−Fc(v)− 1, v is connected to at least one vertex in component C, a contradiction. (ii) Sup-pose u∈ {a, b}. We without loss of generality let u = a, and consider the adjacency between a and b.

L.-M. Shih et al. / Information Processing Letters 106 (2008) 64–69 69

Subcase 1: Suppose that a is not adjacent to b. By the assumption that u and v are separated by deleting a set of vertices Vfcwith|Vfc| = degG_−Fc(u)− 1. Let Vfc be a subset of the neighborhood of u, that is, Vfc⊂ N(u). Since |Vfc| < |N(u)|, vertex u and component C are connected, which is a contradiction.

Subcase 2: Suppose that a is adjacent to b. Let Vfc= N (u)− {b}. Since G − Fcis a conditional faulty graph,

one of the neighbors of b is in C. Then, b is connected to C, which is a contradiction.

Therefore, vertex u and v are still connected with up to deg_G−Fc(u)− 1 vertex faults. By Theorem 1, this implies that each pair of vertices u and v in G− Fc

are connected by min{degG−Fc(u),degG−Fc(v)} vertex-disjoint fault-free paths, where|Fc|  2n − 5. The proof

is complete. 2

Acknowledgements

The authors are grateful to the anonymous referees for a number of comments and suggestions that improve the quality of this paper.

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