最大區域連通度與邊泛迴圈之條件式容錯度之研究

國

立

交

通

大

學

資訊科學與工程研究所

博 士 論 文

最大區域連通度與邊泛迴圈之條件式容錯度研究

Maximally Local-connected and Edge-bipancyclic

Property with Conditional Faults on Interconnection

Networks

研 究 生 ：施倫閔

誌謝

本篇論文能夠順利完成有許多要感謝的人，這些人的付出及給予的幫忙讓我銘感 在心。首先要感謝的是指導老師譚建民教授，您的用心指導與照顧讓我不僅在研 究上精進，也在許多生活及處事上都有不同的觀點及體驗。實驗室的大家：晃哥 、峰哥、韓吉吉、寬哥及阿鋼，謝謝你們一路陪伴這段交大的生活，讓我在研究 學習上可以如此順利。峰哥，因為有你英文能力的幫忙，才能使論文產量如此順 利，在此特別感謝你。 此外，對於論文計畫指導委員：蔡錫鈞教授、陳榮傑教授；校內口試委員曾文貴 教授；校外口試委員王有禮教授、徐力行教授、王炳豐教授及謝孫源教授，非常 感謝您們於口試時給予的建議及指導，讓本論文可以更加完善。 最後，特別要感謝的是我的家人，你們給予我最大的動力。感謝爸媽您們各方面 的支援讓我可以全心全意在課業上而沒有後顧之憂。感謝老婆你的鼓勵及默默的 付出，並把小孩照顧得這麼好，謝謝妳。還有所有在我身邊幫助我的朋友們，謝 謝你們。

最大區域連通度與邊泛迴圈之條件式容錯度研究

研究生：施倫閔

指導教授：譚建民 博士

國立交通大學資訊工程系

摘要

容錯度的問題己經是個相當廣泛討論的主題。在這篇論文當中，我們在幾個連結 網路上研究了一些條件式容錯度的特性。首先，我們討論在n維度的超立方體網 路中，當每個節點都必需要兩個好的節點與其相連時，任何 2n-5 個的邊壞掉的 情形下，對任意一個邊都可以找到長度從 6 到 2n_{的迴圈通過這個邊。並且証明出} 此結果為最佳結果。 接著我們在類超立方體網路中對 Menger 定理做了研究。我們証明在 n 維度的類 超立方體網路中，在壞掉 n-2 個節點之後，對任意一對點 u 跟 v 皆可以找到 min{deg(u), deg(v)}條 vertex-disjoint 的路徑連結這兩個點。若給予每個節點都必 需要兩個好的節點與其相連的條件下，則容錯度可以上升到 2n-5。 最後我們定義了一對一及一對多兩種最大區域連通度的特性。我們証明了在 k+1 正規 MCNs 網路中，在壞掉容錯於小於 k-1 的情形下，都具有一對一及一對多的 最大區域連通度的特性。進一步的我們也討論在 k+1 正規 MCNs 網路中，任意 拔除 f 個點，對於任意獨立兩個大小為 t 的點集合中，可以找到 t 條 vertex-disjoint 的路徑連結這兩個點集合，其中 f 和 t 存在著 f+t

≦

2k 的情形。我們也針對由

transposition tree 生成的 Cayley graph 做相關的研究。在 n-1 正規式 transposition tree 生成的 Cayley 網路中，容錯度在 n-3 的情形下皆有一對一及一對多的區域性 連通度的特性。

關鍵字：連結網路；條件式容錯；容錯；泛迴圈；連通度；Menger 定理；區域 性連通度；MCNs 網路；超立方體網路；類超立方體網路；星狀網路；泡泡性網 路；Cayley 網路

Maximally Local-connected and Edge-bipancyclic Property

with Conditional Faults on Interconnection Networks

Student:

Lun-Min

Shih

Advisor:

Dr.

Jimmy

J.M.

Tan

Department of Computer Science

College of Computer Science

National Chiao Tung University

Abstract

The problem of fault-tolerance has been discussed widely. In this thesis, we study several properties with conditional fault on some interconnection networks. First of all, we show that for any set of faulty edges F of an n-dimensional hypercube Qn with F

≦2n-5, each edge of the faulty hypercube Qn -F lies on a cycle of every even length

from 6 to 2n with each vertex having at least two healthy edges adjacent to it, for

n

≧

3

. Moreover, this result is optimal in the sense that there is a set F of 2n-4 conditional faulty edges in Qn such that Qn -F contains no Hamiltonian cycle.

Second, we study the Menger property on a class of hypercube-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.

The local connectivity of two vertices is defined as the maximum number of internally vertex-disjoint paths between them. Finally, we define two vertices to be maximally local-connected, if the maximum number of internally vertex-disjoint paths between them equals the minimum degree of these two vertices. We prove that a (k+1)-regular Matching Composition Network is maximally local-connected, even if there are at most (k-1) faulty vertices in it. Moreover, we introduce the one-to-many and many-to-many versions of connectivity, and prove that a (k+1)-regular Matching Composition Network is not only (k-1)-fault-tolerant one-to-many maximally

defined subsequently) if f+t=2k. In the same issue, we show that an (n-1)-regular Cayley graph generated by transposition tree is maximally local-connected, even if there are at most (n-3) faulty vertices in it, and prove that it is also (n-1)-fault-tolerant one-to-many maximally local-connected.

Keywords: Interconnection networks; Fault-tolerant; Pancyclic; Conditional faults; Connectivity; Strong Menger connectivity; Local connectivity;

Matching Composition network; Hypercube; Hypercube-like networks; Cayley graphs; Star graph; Bubble-sort graph;

Contents

1 Introductions and Motivations 1

1.1 Basic Terms and Notations . . . 2

1.2 Organization of the Thesis . . . 3

2 Edge-bipancyclicity 5 2.1 The Edge-bipancyclic property . . . 5

2.2 The Hypercube Networks . . . 6

2.3 The Conditional Fault-tolerance of Hypercube Networks . . . 9

2.4 Edge-bipancyclicity of conditional faulty hypercube . . . 12

3 Strong Menger-connectivity 20 3.1 Menger-connectivity and Strong Menger-connectivity . . . 21

3.3 Strong Menger Connectivity on the

Class of Hypercube-like Networks . . . 23 3.4 Strong Menger Connectivity with Conditional

Faults on the Class of Hypercube-like Networks . . . 26

4 Maximal Local-connectivity 33

4.1 Local-connectivity and Maximal Local-connectivity . . . 34 4.2 The Matching Composition Networks . . . 38 4.3 Cayley Graphs Generated by Transposition Trees . . . 42 4.4 Maximal Local-Connectivity on the Matching Composition Networks . . . 43 4.4.1 One-to-One Maximal Local-connectivity . . . 45 4.4.2 One-to-Many Maximal Local-connectivity . . . 49 4.4.3 Many-to-Many Maximal Local-connectivity . . . 52 4.5 Maximal Local-Connectivity on Cayley graphs generated by transposition

trees . . . 54 4.5.1 One-to-One Maximal Local-connectivity . . . 55 4.5.2 One-to-Many Maximal Local-connectivity . . . 57

List of Figures

2.1 An illustration for Theorem 1. . . 14

3.1 The illustration of the proof of Case 1 in Lemma 11. . . 24 3.2 An example showing that an HL₄ is not strongly Menger-connected. . . 26

4.1 An example showing that in a k-connected k-regular graph, there are at most 2k− 2 vertex-disjoint paths between two conditional selected vertex sets. . . 37 4.2 The Star Graph: H ={(12), (13), (14)}. . . 43 4.3 The Bubble-sort Graph: H ={(12), (23), (34)}. . . 44 4.4 An example showing that a (k + 1)-regular MCN is not k-maximally

Chapter 1

Introductions and Motivations

The research about interconnection networks is important for parallel and distributed computer system. Many interconnection network topologies have been proposed in lit-erature for the purpose of connecting a large number of processing elements and the designing of a parallel computing system. There are several requirements in designing a good topology for an interconnection network, such as connectivity and ring embedding. Many related works can be referred in recent research.

In practice, the processors or links in a network may be failure. Since failures are inevitable, fault tolerance is an important issue in multiprocessor systems. The connec-tivity is also related to the reliability and fault tolerance of a network. Many measures on fault tolerance of networks are related to the maximal size of the connected components of networks with faulty vertices/edges. In this dissertation, we consider some measures of conditional faults by restricting that every vertex has at least two fault-free neighbor-ing vertices on some interconnection networks. Under this condition, the fault-tolerant capability is increased.

1.1

Basic Terms and Notations

The architecture of a multiprocessor system is usually modeled as an undirected graph. For the graph definitions and notations we follow [3]. Let G = (V, E) be a graph, we use V (G) and E(G) to denote the vertex set V and the edge set E, respectively. The connectivity of a graph G, written κ(G), is the minimum size of a vertex set S such that G− S is disconnected or has only one vertex. A graph G is k-connected if its connectivity is at least k. In addition, a graph has connectivity k if it is k-connected but not (k + 1)-connected. The degree of a vertex x is the number of edges incident with it. We use deg_G(x), or simply deg(x) if there is no ambiguity, to denote the degree of vertex x in G; and use δ(G) to denote the minimum degree of all the vertices in G. We say that G is maximally connected if κ(G) = δ(G). Let u and v be two distinct vertices, a path P between them is a sequence of adjacent vertices, < u, w₁, w₂, ..., w_k, v >, where w₁, w₂, ..., w_k are distinct ones. The local connectivity between two distinct vertices u and v is the maximum number of internally disjoint u− v paths.

Let G be a graph, and F be a subset of vertices, F ⊂ V (G), the induced subgraph obtained by deleting the vertices of F from G is denoted by G− F . Let u be a vertex, we use N_G(u), or simply N (G) if there is no ambiguity, to denote the set of vertices adjacent to u in G. Let V be a set of vertices, the neighborhood of V is defined as the set N_G(V) ={ 

v∈VNG(v)} − V

_{. A graph G is k-regular if the degree of every vertex in G}

is k, and graph G is triangle-free if there is no cycle of length three. A Hamiltonian cycle is a cycle which includes every vertex of G. A path P is a sequence of adjacent vertices, written asv₀, v₁, ..., v_m. The length of a path P , denoted by l(P ), is the number of edges in P .

1.2

Organization of the Thesis

In the follows, we describe the organization of this thesis. In Chapter 2, we discuss about the edge-bipancyclicity problem with conditional faults on hypercubes. We show that, for up to |F | = 2n − 5 faulty edges, each edge of the faulty hypercube Q_n− F lies on a cycle of every even length from 6 to 2n with each vertex having at least two healthy edges adjacent to it, for n≥ 3.

In Chapter 3, we study the Menger property on a class of hypercube-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.

In Chapter 4, we focus on local connectivity problem. we define two vertices to be maximally local-connected, if the maximum number of internally vertex-disjoint paths between them equals the minimum degree of these two vertices. We prove that a (k + 1)-regular Matching Composition Network is maximally local-connected, even if there are at most (k− 1) faulty vertices in it. Moreover, we introduce the one-to-many and many-to-many versions of connectivity, and prove that a (k + 1)-regular Matching Composition Network is not only (k − 1)-fault-tolerant one-to-many maximally local-connected but also f -fault-tolerant many-to-many t-connected (which will be defined subsequently) if f + t = 2k. Furthermore, we introduce the Cayley graphs generated by transposition tree, and show that an (n− 1)-regular Cayley graph generated by transposition tree is

maximally local-connected, even if there are at most (n−3) faulty vertices in it, and prove that it is also (n− 1)-fault-tolerant one-to-many maximally local-connected. At last, we present our conclusion in Chapter 5.

Chapter 2

Edge-bipancyclicity

2.1

The Edge-bipancyclic property

The ring embedding problem, which deals with all the possible lengths of the cycles in a given graph, is investigated in a lot of interconnection networks. A graph G is pancyclic if it contains a cycle of length l for each l satisfying 3 ≤ l ≤ |V (G)|. The concept of pancyclic graphs is proposed by Bondy [3]. Bipancyclicity is essentially a restriction of the concept of pancyclicity to cycles of even lengths. A bipartite graph is edge-bipancyclic if every edge lies on a cycle of every even length from 4 to|V (G)|. There are some studies concerning ring embedding problem of some interconnection networks [9, 13, 14].

A bipartite graph is k-edge-fault-tolerant edge-bipancyclic if G − F remains edge-bipancyclic for any set of faulty edges F ⊂ E(G) with |F | ≤ k. In this chapter, we discuss that for|F | = 2n − 5 conditional faulty edges, each edges of Q_n− F lies on a cycle of every even length from 6 to 2n, n≥ 4, provided not all edges in F are incident with the same vertex.

2.2

The Hypercube Networks

An n-dimensional hypercube is denoted by Q_n with the vertex set V (Q_n) and the edge set E(Q_n). Each vertex u of Q_n can be distinctly labeled by a n-bit binary strings, u = u_n−1u_n−2...u₁u₀. There is an edge between two vertices if and only if their binary labels differ in exactly one bit position. In addition, we call e a healthy edge when e is fault-free in a graph. Let u and v be two adjacent vertices. If the binary labels of u and v differ in ith position, then the edge between them is said to be in ith dimension and the edge (u, v) is called an ith dimension edge. Let i be a fixed position, we use Q0_n−1 to denote the subgraph of Q_n induced by {u ∈ V (Q_n)|u_i = 0} and Q1_n−1 to denote the subgraph of Q_n induced by {u ∈ V (Q_n)|u_i = 1}. We say that Q_n is decomposed into Q0_n−1 and Q1_n−1 by dimension i, and Q0_n−1 and Q1_n−1 are (n− 1)-dimensional subcube of Q_n induced by the vertices with the ith bit position being 0 and 1 respectively. Q0_n−1 and Q1_n−1 are all isomorphic to Q_n−1. For each vertex u ∈ V (Q0_n−1), there is exactly one vertex in Q1_n−1, denoted by u(1), such that (u, u(1)) ∈ E(Q_n). Conversely, for each u ∈ V (Q1_n−1), there is one vertex in Q0_n−1, denoted by u(0), such that (u, u(0))∈ E(Q_n). Let D_i be the set of all edges with one end in Q0_n−1 and the other in Q1_n−1. These edges are called crossing edges in the ith dimension between Q0_n−1 and Q1_n−1. We also call D_i the set of all ith dimension edges. Consequently, |D_i| = 2n−1 for all 0≤ i ≤ n − 1.

There are some properties on Q_n as follows.

Lemma 1 [16] Q_n is edge-bipancyclic, and is (n-2)-edge-fault-tolerant edge-bipancyclic, for n≥ 3.

for any F ⊂ E(Q₄) with |F | = 3, provided not all the faulty edges in F are incident with the same vertex.

Lemma 3 [29] Any two edges in Q_n are included in a Hamiltonian cycle, for n≥ 2 .

The above lemma can be improved; In addition, we have the following lemmas to simplify our proof.

Lemma 4 Let C₀ =u, P₀, v, u be a cycle in Q0_n−1 with its even length from l₀ to 2n−1, and C₁ =u(1), P₁, v(1), u(1) be a cycle in Q1_n−1 with its even length from l₁ to 2n−1. Then C =u, P₀, v, v(1), P₁, u(1), u is a cycle in Q_n with its even length from l₀+ l₁ to 2n.

Proof. The proof of this lemma is omitted. 2

Lemma 5 Let Q_n be an n-dimensional hypercube, n≥ 2, and let e₁ and e₂ be two edges in the same dimension i. Then there exists another dimension j = i such that decomposing Q_n into Q0_n−1 and Q1_n−1 by dimension j, we have (1) neither e₁ nor e₂ is a crossing edge, (2) not e₁ and e₂ are in the same subcube.

Proof. Let e₁ = (a, b) and e₂ = (s, t) be two edges in the same dimension i. Let a = a_n...a_i...a₁ and s = s_n...s_i...s₁. Then b = a_n...a_i...a₁ and t = s_n...s_i...s₁. Since e₁ = e₂ and n ≥ 2, there exists another dimension j = i, such that a_j = s_j. We decompose Q_n into Q0_n−1 and Q1_n−1 by dimension j. Then, e₁ and e₂ are not crossing edges and are in the different subcubes. 2

Lemma 6 Consider an n-dimensional hypercube Q_n, for n≥ 4. Let e₀, e₁ and e₂ be any three edges in Q_n, there is a cycle C containing e₁ and e₂ in Q_n− {e₀} with the length l(C) = 2n, 2n− 2 and 2n− 4.

Proof. To prove this lemma, we consider the following two cases:

Case 1: Both e₁ and e₂ are in the same dimension, say dimension i. By Lemma 5, we can choose a dimension j such that e₁ and e₂ are in different subcubes. Without loss of generality, we assume that e₁ is in Q0_n−1 and e₂ is in Q1_n−1. We then consider two cases:

1.1: e₀ is not a crossing edge. We assume without loss of generality that e₀ is in Q0_n−1. By Lemma 1, in Q0_n−1−{e₀}, there exists a cycle C₀ of every even length 4≤ l(C₀)≤ 2n−1 going through e₁. Since n ≥ 4, we can choose an edge (u, v) on cycle C₀ such that (u, v) = e₁, and (u(1), v(1))= e₂. By Lemma 3, in Q1_n−1, there exists a cycle C₁ of length 2n−1 going through e₂ and (u(1), v(1)). Thus, the conclusion follows according to Lemma 4.

1.2: e₀ is a crossing edge. By Lemma 1, there exists a C₀ of every even length 4 ≤ l(C₀) ≤ 2n−1 going through e₁ in Q0_n−1. We can choose an edge (u, v) on cycle C₀ such that (u, v) is not adjacent to e₀ and (u, v) = e₁ and (u(1), v(1)) = e₂, since n ≥ 4. By definition, (u(1), v(1)) is an edge in Q1_n−1. By Lemma 3, there exists a Hamiltonian cycle C₁ going through e₂ and (u(1), v(1)) in Q1_n−1. So the conclusion follows according to Lemma 4.

Case 2: e₁ and e₂ are in different dimensions. Suppose that e₀is in the ith dimension. We decompose Q_n into Q0_n−1 and Q1_n−1 by dimension i. Then, e₀ is a crossing edge. Next, we consider two further cases:

2.1: Either e₁ or e₂ is a crossing edge. Without loss of generality, we assume that e₁ is a crossing edge, and e₂ is in Q0_n−1. Let e₁ = (u, u(1)), where u ∈ V (Q0_n−1) and u(1) ∈ V (Q1_n−1). Since n ≥ 4, there is a neighbor of u, say v, such that (u, v) = e₂ and (u, v) is not adjacent to e₀. By Lemma 3, there exists a Hamiltonian cycle C₀ going through (u, v) and e₂ in Q0_n−1. By Lemma 1, in Q1_n−1, there exists a cycle C₁of every even length 4≤ l(C₁)≤ 2n−1 going through (u(1), v(1)). By Lemma 4, the conclusion follows.

2.2: Both e₁ and e₂ are not crossing edge. If e₁ and e₂ are in different subcubes, this subcase is similar to case 1.2, and the proof is omitted. Otherwise, both e₁ and e₂ are in the same subcube. We assume without loss of generality that e₁ and e₂ are in Q0_n−1. By Lemma 3, there exists a Hamiltonian cycle C₀ going through e₁ and e₂ in Q0_n−1, and l(C₀) = 2n−1. Since n≥ 4, there is a third edge (u, v) other than e₁ and e₂ on cycle C₀, and (u, v) is not adjacent to e₀. By Lemma 1, there exists a cycle C₁ of every even length 4≤ l(C₁)≤ 2n−1 going through (u(1), v(1)) in Q1_n−1. By Lemma 4, the conclusion follows. 2

2.3

The Conditional Fault-tolerance of Hypercube

Networks

Chan and Lee [5] considered an injured n-dimensional hypercube where each vertex is incident with at least two healthy edges, and proved that it still contains a Hamiltonian cycle even it has (2n− 5) edge faults. Tsai [26] proved that such an injured hypercube Q_n contains a cycle of every even length from 4 to 2n, even if it has up to (2n− 5) edge faults. Recently, Xu et al. [29] showed that for any set of faulty edges F of Q_n with |F | ≤ n − 1, each edge of Qn− F lies on a cycle of every even length from 6 to 2n, n≥ 4,

provided not all faulty edges are incident with the same vertex. We observe that not all faulty edges are incident with the same vertex is equivalent to stating that each vertex has at least two healthy edges adjacent to it, if|F | ≤ n − 1. In this chapter, we consider a set of faulty edges satisfying the condition that each vertex of Q_n− F is incident with at least two healthy edges. Such a set of faulty edges F is called a set of conditional faulty edges.

To prove our result, we need some preliminary lemmas.

Lemma 7 Consider an n-dimensional hypercube Q_n, for n≥ 4. Let F be a set of condi-tional faulty edges with |F | = 2n − 5. There are at most two vertices in Q_n incident with (n-2) faulty edges.

Proof. If there are three vertices in Q_n incident with (n− 2) faulty edges, the number of faulty edge F is at least 3n− 8. However, (3n − 8) > (2n − 5) for all n ≥ 4 which is a contradiction. 2 Let F be a set of faulty edges of Q_n. Suppose that we decompose Q_n into Q0_n−1 and Q1_n−1 by dimension j, and let F_L = F ∩ E(Q0_n−1), F_R = F ∩ E(Q1_n−1). Suppose that F is a set of conditional faulty edges of Q_n. If we arbitrarily decompose Q_n into Q0_n−1 and Q1_n−1 by a dimension, F_L and F_R may not be conditional faulty edges in Q0_n−1 and Q1_n−1 respectively. However, we will show that it is always possible to find some suitable dimension such that decomposing by this dimension, both F_L and F_R are conditional faulty sets in Q0_n−1 and Q1_n−1 respectively.

Lemma 8 Consider an n-dimensional hypercube Q_n, n≥ 4. Let F be a set of conditional faulty edges with |F | = 2n − 5. If there are two vertices x and y both incident with n-2

faulty edges, then x and y are adjacent in Q_n and the edge (x,y) is a faulty edge. Suppose that (x,y) is in dimension j. Then decomposing Q_n into Q0_n−1 and Q1_n−1 by dimension j, both F_L and F_R are sets of conditional faulty edges in Q0_n−1 and Q1_n−1 respectively. Moreover, |F_L| ≤ 2n − 6 and |F_R| ≤ 2n − 6.

Proof. If there are two vertices x and y in Q_nincident with (n−2) faulty edges, then these two vertices are connected by a faulty edge. Otherwise, |F | = 2(n − 2) = 2n − 4 > 2n − 5 which is a contradiction. Suppose the edge (x, y) is in dimension j, we decompose Q_n into two subcubes. It is clearly that each vertex in Q0_n−1 and Q1_n−1 is still incident with at least two healthy edges, and both F_L and F_R are conditional faulty edges in Q0_n−1 and Q1_n−1 respectively. Then, |F_L| = |F_R| = n − 3 ≤ 2n − 6, for n ≥ 4. 2

Lemma 9 Consider an n-dimensional hypercube Q_n, for n ≥ 4. Let F be a set of con-ditional faulty edges with |F | = 2n − 5. Suppose that there exists exactly one vertex x having (n-2) faulty edges incident with it. Since n− 2 ≥ 2, let e₁ and e₂ be two faulty edges incident with x, and let e₁ and e₂ be jth and kth dimension edges respectively. Then decomposing Q_n into Q0_n−1 and Q1_n−1 by either one of these two dimensions j and k, F_L and F_R are still sets of conditional faulty edges in Q0_n−1 and Q1_n−1 respectively. Moreover, |FL| ≤ 2n − 6 and |FR| ≤ 2n − 6.

Proof. If there exists only one vertex x having (n−2) faulty edges incident with it, there

are at least two faulty edges e₁ and e₂ incident with it, since n ≥ 4. Obviously, these two faulty edges are in different dimensions. Without loss of generality, we may assume that e₁ is in dimension j and e₂ is in dimension k, for j = k. We can decompose Q_n into Q0_n−1 and Q1_n−1 by either jth or kth dimension, and either e₁ or e₂ is a crossing edge.

Therefore, each vertex in these two subcubes is incident with at least two healthy edges and |F_L| ≤ 2n − 6 and |F_R| ≤ 2n − 6. 2

Lemma 10 Let Q_n be an n-dimensional hypercube, F be a set of faulty edges with|F | ≥ 2, and e be a healthy edge, n ≥ 2. Then there exists a dimension j, decomposing Q_n into Q0_n−1 and Q1_n−1 by this dimension, such that e is not a crossing edge and not all the faulty edges are in the same subcube.

Proof. Suppose that e = (u, v) is in dimension i. If there is a faulty edge f not in

dimension i, say in dimension j. We decompose Q_n into Q0_n−1 and Q1_n−1 by dimension j. Then f is a crossing edge but e is not, and all the faulty edges are not in the same subcube. Otherwise, all the faulty edges are in the same dimension i as e is in. We now choose any two faulty edges f₁ and f₂ in F . By Lemma 5, Q_n can be decomposed into Q0_n−1 and Q1_n−1 by some dimension j = i such that edges f₁ and f₂ are not in the same subcube, and e is not a crossing edge. 2

2.4

Edge-bipancyclicity of conditional faulty

hyper-cube

In this section, we consider a set of faulty edges satisfying the condition that each vertex of Q_n−F is incident with at least two healthy edges. Such a set of faulty edges F is called a set of conditional faulty edges and Q_n− F is called a conditional faulty hypercube. We find that under this condition, the number of faulty edges can be much greater and the same result still holds. We show that, for up to |F | = 2n − 5 conditional faulty edges, each edge of a faulty hypercube Q_n− F lies on a cycle of every even length from 6 to 2n, for n≥ 3. We observe that, if |F | < 2n − 5, we may arbitrarily delete some more edges

to make a faulty edge set F ⊇ F and |F| = 2n − 5. If our result holds for F, it holds for F . From now on, we shall assume |F | = 2n − 5.

The above result is optimal in the sense that the result can not be guaranteed, if there are 2n− 4 conditional faulty edges. For example, take a cycle of length four in Q_n, let u1, u2, u3, u4 be the consecutive vertices on this cycle. Suppose that all the (n−2) edges

incident with vertex u₁ (respectively vertex u₃) are faulty except those two edges on the four cycle are healthy. There are 2(n− 2) conditional faulty edges. Then there does not exist a Hamiltonian cycle in this faulty Q_n, for n≥ 3.

We now prove our main result.

Theorem 1 Let Q_n be an n-dimensional hypercube, and F be a set of conditional faulty edges with |F | ≤ 2n − 5. Then each edge of the conditional faulty hypercube Q_n− F lies on a cycle of every even length from 6 to 2n, for n≥ 3.

Proof. We prove this theorem by induction on n. For n = 3, since 2n− 5 = n − 2, by

Lemma 1, the result is true. For n = 4, 2n− 5 = n − 1, by Lemma 2, the result holds. Assume the theorem holds for n− 1, for some n ≥ 5, we shall show that it is true for n.

As we mentioned before, we may assume |F | = 2n − 5. Let e = (u, v) be an edge in Q_n − F . We shall find a cycle of every even length from 6 to 2n passing through e in Q_n− F . Assume that e is an ith dimension edge, e ∈ D_i, for some i ∈ {1, 2, ..., n}. The proof is divided into three major cases:

Case 1: There are two vertices x and y in Q_n incident with (n− 2) faulty edges. By Lemma 8, (x, y) is an edge in Q_n and is a faulty edge. We denote this edge by e_f. Suppose

(a) (b) (c) (d) (e) (f) ¡› ¡› ¡› ¡› ef ef e u v w w(1) P₀ P₁ e s t s(1) t(1) P0 P₁ ¡› ¡› ¡› ¡› e_f s t s(1) t(1) e P₀ P1 w v e u w(1) v(1) e u(1) v(1) u v _e s t s(1) t(1) P₀ P₁ Q0 n-1 Q 1 n-1 Q 0 n-1 Q 1 n-1 Q0_n-1 Q1_n-1 Q0 n-1 Q 1 n-1 Q0 n-1 Q 1 n-1 Q0 n-1 Q 1 n-1 e_f ef ef

Figure 2.1: An illustration for Theorem 1.

that e_f is a jth dimension edge. We decompose Q_n into Q0_n−1 and Q1_n−1 by dimension j. We then consider two further cases:

1.1 e_f = (x, y) and e = (u, v) are in the same dimension. Thus, j = i and e_f ∈ D_i. (Fig. 2.1-(a)) In this case, e is an edge crossing Q0_n−1and Q1_n−1. Without loss of generality, assume that u∈ V (Q0_n−1) and v ∈ V (Q1_n−1). Since n≥ 5, u has a neighboring vertex w ∈ V (Q0_n−1), by the definition of hypercube, w(1)is a neighbor of v such that the edge (w, w(1)) is a healthy edge and (w, w(1)) is a crossing edge between Q0_n−1 and Q1_n−1. By lemma 1, there exists a cycle C₀in Q0_n−1−F_Lpassing through (u, w) of every even length 4≤ l(C₀)≤ 2n−1 and a cycle C₁ in Q1_n−1−F_R going through (v, w(1)) of every even length 4≤ l(C₁)≤ 2n−1. We write C₀ as u, P₀, w, u, and C₁ as v, P₁, w(1), v. Thus, u, P₀, w, w(1), v, u is a cycle of length 6 with l(P₀) = 3. By Lemma 4, u, P₀, w, w(1), P₁, v, u can form a cycle of every even length from 8 to 2n through e in Q_n− F .

1.2 e_f and e are in different dimensions. Thus, j = i and e_f ∈ D/ _i. (Fig 2.1-(b)) In this case, e is in Q0_n−1 or Q1_n−1. Without loss of generality, we may assume that e∈ E(Q0_n−1).

By Lemma 1, there exists a cycle C in Q0_n−1− F_L going through the edge e of every even length l, 6 ≤ l ≤ 2n−1. Let C₀ be a cycle of length 2n−1− 2 or 2n−1 passing through e in Q0_n−1− F_L. Since n≥ 5, there exists an edge (s, t) on C₀ such that neither s nor t is adjacent to e_f and (s, t) = e. By definition, (s(1), t(1)) is an edge in Q1_n−1, and (s, s(1)), (t, t(1)) are healthy edges. By Lemma 1, there exists a cycle C₁ in Q1_n−1 − F_R through (s(1), t(1)) of every even length 4≤ l(C₁) ≤ 2n−1. Thus, the conclusion follows according to Lemma 4.

Case 2: There is exactly one vertex in Q_n incident with (n− 2) faulty edges. Let x be the vertex having (n− 2) faulty edges incident with it. Let f₁ and f₂ be two faulty edges incident with x, so f₁ and f₂ are in different dimensions j and k. By Lemma 9, decomposing Q_n into Q0_n−1 and Q1_n−1 by either jth or kth dimension, both F_L = F ∩ E(Q0_n−1) and F_R = F ∩ E(Q1_n−1) are sets of conditional faulty edges in Q0_n−1 and Q1_n−1 respectively. Between dimension j and k, we choose one to decompose Q_n into Q0_n−1 and Q1_n−1, say dimension j, such that the required edge e is not a crossing edge. Therefore, there is an faulty edge crossing Q0_n−1 and Q1_n−1, we denote this edge by e_f, and e_f ∈ F ∩ D_j is incident with x. Without loss of generality, we may assume that x∈ V (Q0_n−1).

2.1: Suppose|F_L| ≤ 2n−7 and |F_R| ≤ 2n−7. (Fig. 2.1-(c)) Without loss of generality,

we further assume that e ∈ E(Q0_n−1). By induction hypothesis, there exists a cycle C in Q0_n−1− F_L of every even length 6≤ l(C) ≤ 2n−1 passing through e. Let C₀ be a cycle of length 2n−1− 4 ≤ l(C₀)≤ 2n−1 through e in Q_n−10 − F_L. Since |C₀− e| ≥ 2n−1− 4 − 1 > 2(2n− 5) ≥ 2|F ∩ D_j|, for all n ≥ 5. There exists an edge (s, t) on C₀ such that (s, t) is not e, and both (s, s(1)) and (t, t(1)) are healthy edges. By induction hypothesis, there exists a cycle C₁ in Q1_n−1 − F_R of every even length 6 ≤ l(C₁) ≤ 2n−1 passing through

(s(1), t(1)). By Lemma 4, the conclusion follows.

2.2: |F_L| = 2n − 6. In this case, |F ∩ D_j| = 1 and |F ∩ E(Q1_n−1)| = |F_R| = 0.

2.2.1: e is in subcube Q0_n−1. To find a cycle of length 6 passing through e = (u, v), we discuss the case that whether e is incident with x or not. If e is incident with x, without loss of generality, we assume that u = x. (Fig. 2.1-(d)) Thus, (v, v(1)) is a healthy edge. Since F_L is a set of conditional faulty edges in Q0_n−1, vertex u = x has two healthy edges incident with it. Let w be a neighbor of u in Q0_n−1 such that (w, u) and (w, w(1)) are healthy edges and w= v. Thus, u, v, v(1), u(1), w(1), w, u is a cycle of length 6 in Q_n− F . Otherwise, e is not incident with x, then (u, u(1)) and (v, v(1)) are healthy edges. (Fig. 2.1-(e)) By Lemma 1, there exists a cycle C₁ =u(1), P₁, v(1), u(1) of length four in Q1_n−1 through the edge (u(1), v(1)). Thus, u, u(1), P₁, v(1), v, u is a cycle of length 6 in Q_n− F , where l(P₁) = 3.

Let e₁ be a faulty edge in Q0_n−1 that is not adjacent to e_f. Though e₁ is a faulty edge, we treat it as a healthy edge temporarily, then the total number of faulty edge in Q0_n−1 is 2n− 7. By induction hypothesis, there exists a cycle C₀ of every even length 6≤ l(C₀)≤ 2n−1 going through e in Q0_n−1− {F_L− {e₁}}. If C₀ passes e₁, we choose e₁ , or else, we choose any one edge other then e on C₀ which is not adjacent to e_f. Let the chosen edge be denoted by (s, t). We write cycle C₀ ass, P₀, t, s. Since |F ∩D_j| = 1 and |FR| = 0, (s, s(1)), (t, t(1)) and (s(1), t(1)) are all healthy edges. Thus, s, P0, t, t(1), s(1), s

is a cycle of length 8 in Q_n− F if l(P₀) = 5. Suppose that 10≤ l ≤ 2n and l is even. By Lemma 1, in Q1_n−1, there exists a cycle C₃ of length 4 ≤ l(C₃) ≤ 2n−1 passing through (s(1), t(1)). We write C₃ as s(1), P₃, t(1), s(1). By Lemma 4, s, P₀, t, t(1), P₃, s(1), s, t is a cycle of length l through e in Q_n − F .

2.2.2: e is in subcube Q1_n−1. (Fig. 2.1-(f)) By Lemma 1, there exists a cycle C of every even length 4≤ l ≤ 2n−1 passing through e in Q1_n−1. Suppose that 2n−1+ 2≤ l ≤ 2n and l is even. Since F_L is a set of conditional faulty edges, there are at most (n− 3) faulty edges adjacent to e_f in Q0_n−1. For n≥ 5, n−3 ≥ 2, we can choose a faulty edge e₂ = (s, t) in Q0_n−1such that e₂ is not adjacent to e_f and (s(1), t(1)) is not e. Treating the edge e₂as a healthy edge, by induction hypothesis, there exists a cycle C₀ of length 6≤ l(C₀)≤ 2n−1 going through e₂ in Q0_n−1− {F_L− {e₂}}. We observe that (s, s(1)) and (t, t(1)) are healthy edges. By Lemma 6, there exists a cycle C₁ of every length 2n−1 − 4, 2n−1− 2, or 2n−1 through (s(1), t(1)) and e in Q1_n−1. By Lemma 4, the conclusion follows.

Case 3: Every vertex in Q_n is incident with at most (n− 3) faulty edges. In this case, suppose that e = (u, v) is in dimension i. By Lemma 10, Q_n can be decomposed into Q0_n−1 and Q1_n−1 by a dimension j different from i such that e is not a crossing edge and not all the faulty edges are in the same subcube. Then |F_L| ≤ 2n − 6 and |F_R| ≤ 2n − 6. Next, we consider two further cases:

3.1: At least one faulty edge is a jth dimension edge. Thus, |F ∩ D_j| = 0.

We then consider two cases: (a)|F_L| ≤ 2n − 7 and |F_R| ≤ 2n − 7, and (b)|F_L| = 2n − 6 or |F_R| = 2n − 6. The proof of this subcase is exactly the same as that of case 2.

3.2: None of the faulty edges is a jth dimension edge. Thus, |F ∩ D_j| = 0.

3.2.1: |F_L| ≤ 2n − 7 and |F_R| ≤ 2n − 7. Without loss of generality, we may assume

that e ∈ E(Q0_n−1). By induction hypothesis, there exists a cycle C of every even length 6≤ l(C) ≤ 2n−1 in Q0_n−1− F_L passing through e. Let C₀ be a cycle of every even length 2n−1 − 4 ≤ l(C₀) ≤ 2n−1 going through e in Q0_n−1 − F_L. There exists an edge (s, t) other than e in C₀. Since |F ∩ D_j| = 0, (s, s(1)) and (t, t(1)) are healthy edges. We write

C₀ as s, P₀, t, s. By induction hypothesis, there exists a cycle C₁ of every even length 6 ≤ l(C₁) ≤ 2n−1 in Q1_n−1 − {F_R− (s(1), t(1))} through (s(1), t(1)). Thus, the conclusion follows according to Lemma 4.

3.2.2: Suppose |F_L| = 2n − 6 or |F_R| = 2n − 6, say the former case. In this case,

|FR| = 1. We then consider two cases: (a) e is in subcube Q0n−1, and (b) e is in subcube

Q1_n−1.

(a) e = (u, v) is in subcube Q0_n−1. Since |F ∩ D_j| = 0, both (u, u(1)) and (v, v(1)) are healthy edges. Let l be an even number with 6 ≤ l ≤ 2n−1. By Lemma 1, there exists a cycle C₁of every even length from 4 to 2n−1 passing through (u(1), v(1)) in Q1_n−1− {FR− (u(1), v(1))}. We write C1 as u(1), P1, v(1), u(1). No matter (u(1), v(1)) is healthy or

not, u, u(1), P₁, v(1), v, u forms a cycle of length l through e in Q_n − F . Suppose that 2n−1+ 2≤ l ≤ 2n. Let e₁ be a faulty edge in Q0_n−1. We may treat e₁ as a healthy edges temporarily. By induction hypothesis, there exists a cycle C₀ of length 6≤ l(C₀)≤ 2n−1 going through e in Q0_n−1 − {F_L − {e₁}}. If C₀ passes the edge e₁, we choose e₁ to be deleted. Otherwise, we choose another edge other than e on cycle C₀. Let the chosen edge be denoted by (s, t). We write the cycle C₀ as s, P₀, t, s. Treating (s(1), t(1)) as a healthy edge, by Lemma 1, there exists a cycle C₃ of every even length from 4 to 2n−1 passing through (s(1), t(1)) in Q1_n−1 − {F_R − (s(1), t(1))}. By Lemma 4, the conclusion follows.

(b): e is in subcube Q1_n−1. Let e₁ be the only faulty edge in Q1_n−1. By Lemma 1, there exists a cycle C of every even length from 6 to 2n−1 through e in Q1_n−1 − {e₁}. Suppose that 2n−1+ 2≤ l ≤ 2n, and l is even. Let e₀ = (s, t) be a faulty edge in Q0_n−1 such that (s(1), t(1)) = e and (s(1), t(1)) = e₁. By induction hypothesis, there exists a cycle C₀ of

length 6≤ l(C₀)≤ 2n−1 in Q0_n−1− {F_L− {e₀}} going through e₀. If (s(1), t(1)) = e₁, treat e₁ as a healthy edge temporarily, by Lemma 6, there exists a cycle C₁ of length 2n−1− 4, 2n−1− 2, or 2n−1 respectively going through both (s(1), t(1)) and e in Q1_n−1. By Lemma 4, the conclusion follows. Otherwise, if (s(1), t(1))= e₁, by Lemma 6, there exists a cycle C₃ of length 2n−1, 2n−1− 2, or 2n−1− 4, respectively, going through both e and (s(1), t(1)) in Q1_n−1− {e₁}. Thus, the conclusion follows according to Lemma 4. 2

Chapter 3

Strong Menger-connectivity

The architecture of an interconnection network is usually denoted as an undirected graph G. Among all fundamental properties for interconnection networks, the (vertex) connec-tivity is a major parameter widely discussed for the connection status of networks. A basic definition of the connectivity of a graph G is defined as the minimum number of vertices whose removal from G produces a disconnected graph. In contrast to this con-cept, Menger [17] 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 chapter, we study the Menger property on a class of hypercube-like networks [27], 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 [18, 19, 20]. 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.

3.1

Menger-connectivity and

Strong Menger-connectivity

In this section, we discuss the strong Menger-connected property. A classical theorem about connectivity was provided by Menger as follows.

Theorem 2 [17] Let x and y be two nonadjacent vertices of a graph G. The minimum

size of an x,y-cut equals the maximum number of pairwise internally disjoint x,y-paths.

Following this theorem, OH et al. [19] gave a definition to extend the Menger’s The-orem.

Definition 1 [19] 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_G−F(u), deg_G−F(v)} vertex-disjoint fault-free paths in G − F , where deg_G−F(u) and deg_G−F(v) are the degree of u and v in G− F , respectively.

By Definition 1, OH et al. [18, 19, 20] showed that an n-dimensional star graph S_n (respectively, an n-dimensional hypercube Q_n)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_G−F(u), deg_G−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.

3.2

The Class of Hypercube-like Networks

Let G₀ = (V₀, E₀) and G₁ = (V₁, E₁) be two disjoint 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) ∈ V₁ and φ : V₀ → V₁ is a bijection}. We use G₀ ⊕_M G₁ to denote 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 topologies in interconnection networks. Several variants of hypercubes are proposed by twisting some pairs of links in hypercubes, including twisted cubes [1, 12], M¨obius cubes [8], and crossed cubes [11], to name a few. To make a unified study on these variants, Vaidya et al. [27] 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 = K₁, where K₁ is a trivial graph in the sense that it has only one vertex; and (2)G ∈ HL_n if and only if G = G₀⊕_M G₁ for some G₀, G₁ ∈ HL_n−1. By the definitions above if G is a graph in HL_n, then G is a composition of G₀⊕_MG₁ with both G₀ and G₁ in HL_n−1, n≥ 1. Each vertex in G₀ has exactly one neighbor in G₁.

It is known that the connectivity of an n-dimensional hypercube-like network HL_n is n [27]. 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 2n− 5 vertex faults.

3.3

Strong Menger Connectivity on the

Class of Hypercube-like Networks

There are some preliminaries on the class of Hypercube-like Networks.

Lemma 11 Let G∈ HL_n be an n-dimensional hypercube-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 trivially 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 ⊕_M G₁, and G₀, G₁ ∈ HL_n−1. Let S be a set of vertices with |S| ≤ 2n − 3, for n ≥ 3, and let S₀ and S₁ be subsets of set S in G₀ and 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 G₀ is (n− 1)-connected, G₀− S₀ is 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− S₁ are 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. 3.1.)

G₀ G₁ S₀

S₁ C₀

Figure 3.1: The illustration of the proof of Case 1 in Lemma 11.

Since 2≤ |S₀| ≤ |S₁| ≤ 2n − 5, so |S₀| ≤ n − 2 and n ≥ 4. By induction hypothesis, there exists a connected component C₁ in G₁− S₁, and |V (C₁)| ≥ 2n−1− |S₁| − 1. Since the connectivity of G₀ is n− 1 and |S₀| ≤ n − 2, G₀− S₀ is 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 11, we have the following corollary.

Corollary 1 Let G be an n-dimensional hypercube-like network, n ≥ 2, and let V be a set of vertices in G with |V| = 2. Then |N(V)| ≥ 2n − 2.

In the following, we show that with up to n− 2 vertex faults, an n-dimensional hypercube-like network has strongly Menger-connected property. Referring to the relative study proposed by OH et al. [18], the strong Menger connectivity of regular hypercube networks has been proved. Here we provide a significantly simpler proof for the general hypercube-like networks.

We now prove our main result.

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 remaining 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 generality that deg_G−F(u)≤ deg_G−F(v), so min{deg_G−F(u), deg_G−F(v)} = deg_G−F(u). We now show that u is connected to v if the number of vertices deleted is smaller than deg_G−F(u)− 1 in G − F . By Theorem 2, this implies that each pair of vertices u and v in G− F are connected by deg_G−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 V_f, where |V_f| ≤ deg_G−F(u)− 1. As a consequence, |V_f| ≤ n − 1 because of deg_G−F(u) ≤ deg(u) ≤ n. Then, the summation of the cardinality of these two sets F and V_f is |F | + |V_f| ≤ 2n − 3. Let S = F ∪ V_f. By Lemma 11, 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 disconnected. 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 V_f must be the neighborhood of u and |V_f| = deg_G−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 .

3.4

Strong Menger Connectivity with Conditional

Faults on the Class of Hypercube-like Networks

As proved in the previous section, an n-dimensional hypercube-like network with at most n− 2 faulty vertices is strongly Menger-connected. But the result can 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 neighbors 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-dimensional 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. 3.2. The remaining degrees of nodes u and v are both four, with three vertices deleted as indicated in the graph. But the number of vertex-disjoint paths between u and v is three.) So we can only expect the result holds for n≥ 5.

u

X X X

v

Figure 3.2: An example showing that an HL₄ is not strongly Menger-connected. 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 12 Let V be a set of vertices in a 4-dimensional hypercube-like network with |V_{| = 3. Then, |N(V}_{)| ≥ 7.}

Proof. Let G be a 4-dimensional hypercube-like network. G is a composition of two

3-dimensional hypercube-like networks G₀ and G₁, G = G₀⊕_MG₁, for a matching operation ⊕M. Without loss of generality, let V be a subset of V (G) containing three vertices

{x, y, z}. If x, y, z are all in G0, by Lemma 11,{x, y, z} has at least 4 neighboring vertices

in G₀. Besides,{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 11, {x, y} has at least 4 neighboring vertices in G₀. In addition,{z} has 3 neighboring vertices in G₁. Then,|N({x, y, z})| ≥ 4+3 = 7. 2

Lemma 13 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 network, G₀, G₁ ∈ HL₄, and G = G₀⊕_MG₁, for a matching operation⊕_M. Let S be a set of vertices with|S| ≤ 3n −6 = 9, for n = 5, and let S₀ and S₁ be subsets of S in G₀ and 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 G₀is (n−1)-connected, G₀−S₀ is connected, for n = 4. So G₀−S₀ has only one connected component C₀ with |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 C is 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−S₀ has only one connected component C₀with|V (C₀)| = 24−S₀. Then, all vertices in G₁are connected to component C₀, except for the three vertices in G₁ adjacent to the vertices in S₀. Since |S₁| ≤ 6 and by Lemma 12, 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 11, there exists 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 13, the general case for n≥ 5 is stated as follows.

with |S| ≤ 3n − 6, for 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 13, 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∈ HL_n−1, and G = G₀⊕_M G₁, for some matching 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 connected, for n≥ 6. Let C₀= G₀−S₀, C₀ is 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 C₀and C₁in G₀−S₀and 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 number 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 G₀ is (n− 1)-connected, G₀ − S₀ is connected. G₀ − S₀ has only one connected component C₀ and |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− |S₁| − 1 ≥ 2n− |S| − 2. 2

Corollary 2 Let G be an n-dimensional hypercube-like network, n ≥ 5, and let V be a set of vertices in G with |V| = 3. Then |N(V)| ≥ 3n − 5.

For the next theorem, we define a set of vertices F_c in graph G to be a conditional faulty vertex set if, in the induced subgraph G−F_c, every vertex has at least two fault-free neighboring vertices. We also call the subgraph G− F_c a conditional faulty graph.

Theorem 4 Consider an n-dimensional hypercube-like network G∈ HL_n, for n≥ 5. Let F_c be a set of conditional faulty vertices with |F_c| ≤ 2n − 5. Then each pair of vertices u and v in G− F_c are connected by min{deg_G−Fc(u), deg_G−Fc(v)} vertex-disjoint fault-free paths, where deg_G−Fc(u) and deg_G−Fc(v) are the degree of u and v in G− F_c, respectively.

Proof. Without loss of generality, we assume deg_G−Fc(u) ≤ deg_G−Fc(v), and therefore min{deg_G−Fc(u) , deg_G−Fc(v)} = deg_G−Fc(u). We want to prove that each pair of vertices u and v in G− F_c are connected by deg_G−Fc(u) vertex-disjoint fault-free paths, for |F_c| ≤ 2n− 5. We are going to show that u is connected to v if the number of vertices deleted is smaller than deg_G−Fc(u)− 1 in G − F_c, where |F_c| ≤ 2n − 5.

Suppose on the contrary that u and v are separated by deleting a set of vertices V_fc, where |V_fc| ≤ deg_G−Fc(u)− 1. By deg_G−Fc(u) ≤ deg(u) ≤ n, we have |V_fc| ≤ n − 1. We

sum up the cardinality of these two sets F_c and V_fc. Since|F_c| ≤ 2n − 5 and |V_fc| ≤ n − 1, then |F_c| + |V_fc| ≤ 3n − 6. Let S = F_c ∪ V_fc. By Lemma 14, 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 assumption 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 |V_fc| ≤ deg_G−Fc(v)− 1, v is connected to at least one vertex in component C, a contradiction. (ii) Suppose u∈ {a, b}. We without loss of generality let u = a, and consider the adjacency between a and b.

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 V_fc with |V_fc| = deg_G−Fc(u)− 1. Let V_fc be a subset of the neighborhood of u, that is, V_fc ⊂ N(u). Since |V_fc| < |N(u)|, vertex u and component C are connected, which is a contradiction.

Subcase 2: Suppose that a is adjacent to b. Let V_fc = N (u)− {b}. Since G − F_c is a conditional faulty graph, one of the neighbors of b is in C. Then, b is connected to C, which is a contradiction.

By Theorem 2, this implies that each pair of vertices u and v in G− F_c are connected by min{deg_G−Fc(u), deg_G−Fc(v)} vertex-disjoint fault-free paths, where |F_c| ≤ 2n − 5. The proof is complete. 2

Chapter 4

Maximal Local-connectivity

One of the central issue in various interconnection networks is studying the value of connectivity. A basic definition of the connectivity of a graph is defined as the minimum number of vertices whose removal results in a disconnected or trivial graph. In contrast to this concept, Menger [17] provided a local point of view, and define the connectivity of any two vertices as the minimum number of vertex-disjoint paths between them. Following this concept, Volkman [28] discussed some issues on it, Oh et al. [19][20] and Shih et al. [23][24] investigated some related properties on the star graph and the class of hypercube-like networks, respectively.

In this chapter, we define two vertices to be maximally local-connected, if the maxi-mum number of internally vertex-disjoint paths between them equals the minimaxi-mum degree of these two vertices. Moreover, we introduce the one-to-many and many-to-many versions of connectivity.

4.1

Local-connectivity and Maximal Local-connectivity

The local connectivity of two vertices is defined as the maximum number of internally vertex-disjoint paths between them. A pair of vertices x and y is maximally local-connected if the local connectivity of x and y equals min{deg(x),deg(y)}, and a graph G is maximally local-connected if every pair of vertices in G are maximally local-connected.

Now we give the definition of a graph to be f -fault-tolerant maximally local-connected.

Definition 2 A graph G is fault-tolerant maximally local-connected, abbreviated as

f-maximally local-connected, if for a set of faulty vertices F , |F | ≤ f, each pair of vertices x, y of G−F are connected by min{deg_G−F(x), deg_G−F(y)} vertex-disjoint fault-free paths, where deg_G−F(x) and deg_G−F(y) are the degrees of x and y in G− F , respectively.

In the previous definition, we discuss the maximal local-connectivity, indicating that for every pair of vertices in a graph with a reasonable number of faulty vertices, there is an amount of vertex-disjoint paths between them, where the amount depends on the minimum remaining degree of the two vertices. Now we shall extend this concept to a “one-to-many” version. In this approach, we consider a vertex (as a source) and a set of vertices (as destinations). Under some constraints we prove that there exists a set of disjoint paths between the source and the destinations.

A classical theorem about the one-to-many connectivity was provided by Dirac[10] as follows.

Definition 3 [10] Given a vertex x and a set U of vertices, an x, U -fan of size k is a set

We note that the cardinality of U is necessarily greater than k, |U| ≥ k.

Theorem 5 [10] A graph is k-connected if and only if it has at least k + 1 vertices and,

for every choice of x, U with |U| ≥ k, it has an x, U-fan of size k.

We then give the definition of a graph to be one-to-many f-fault-tolerant maximally local-connected.

Let G be a graph and F be a set of faulty vertices. In graph G− F , pick an arbitrary vertex x (as a source) and a set U of vertices (as destinations) with |U| = t, x /∈ U. We want to find a set of t paths from x to U such that each pair of them share only the vertex x. In order to do so, the set U must satisfy some necessary conditions: (i) the cardinality of U is not greater than the remaining degree of x, that is, |U| ≤ deg_G−F(x), and (ii) the set U cannot contain any vertex and all its neighbors, that is, {v} ∪ N_G−F(v) is not contained in U for each v ∈ U.

We call a set of vertices U in G− F satisfying the above two conditions a conditional terminal set with respect to x and F , abbreviated as a conditional terminal set if there is no ambiguity. As a short remark, we note that U ⊆ V (G − F ), x /∈ U, |U| ≤ deg_G−F(x), and U ⊇ {v} ∪ N_G−F(v) for each v ∈ U.

Definition 4 A graph G is one-to-many f-fault-tolerant maximally local-connected,

ab-breviated as one-to-many f -maximally local-connected, if given any set of faulty vertices F with |F | ≤ f and x ∈ G − F , let t ≤ deg_G−F(x), there is a set of t paths from x to U such that each pair of them share only the vertex x, for each conditional terminal set U with |U| = t.