Edge fault tolerance of super edge connectivity for three families of interconnection networks

doi:10.1016/j.ins.2011.11.006

Information Sciences

Volume 188, 1 April 2012, Pages 260-268

https://doi.org/10.1016/j.ins.2011.11.006 Get rights and content

Abstract

Let G = (V, E) be a connected graph. G is said to be super edge connected (or super-λ for short) if every minimum edge cut of G isolates one of the vertex of G. A graph G is called m-super-λ if for any edge set S ⊆ E(G) with ∣S∣ ⩽ m, G − S is still super-λ. The maximum cardinality of m-super-λ is called the edge fault tolerance of super edge connectivity of G. In this paper, we discuss the edge fault tolerance of super edge connectivity of three families of interconnection networks.

Introduction

We use Bondy and Murty [3] for terminology and notation not defined here and only consider finite simple undirected graphs. Let G = (V, E) be a connected graph. For v ∈ V(G), the degree of v, written by d(v), is the number of edges incident with v. Let δ(G) = min {d(v)∣v ∈ V(G)} and it is called the minimum degree of G. For a subset S of V(G), G[S] is the subgraph of G induced by S. An edge subset T ⊆ E(G) is an edge cut if G − T is disconnected. The edge connectivity, denoted by λ, is the minimum cardinality of the set of all edge cuts of G.

It is well known that the edge connectivity λ is an important measurement for the fault tolerance of networks. In general, the larger λ is, the more reliable a network is. Obviously, λ ⩽ δ(G). In [2], Bauer et al. defined the so-called super-λ graphs. A graph G is said to be super edge-connected (in short, super-λ) if every minimum edge cut is the set of edges incident with some vertex of G. There are much research on super-λ, the reader is referred to [5], [11], [13], [16], [18] and the references therein.

In [8], [9], Esfahanian and Hakimi proposed the concept of restricted edge connectivity of graphs which generalized the concept of super-λ. Then Fábrega and Fiol [10] introduced the k-restricted edge connectivity of interconnection networks. Let G be a graph. An edge set S ⊂ E is said to be a k-restricted edge cut if G − S is disconnected and there are no components whose cardinalities are smaller than k in G − S. The minimum cardinality of k-restricted edge cut of G is called k-restricted edge connectivity of G, denoted by λ_k(G). k-restricted edge connectivity is another important parameter in measuring the reliability and fault tolerance of large interconnection networks. In particular, estimating the bound for λ_k(G) is of great interest, and many results have been obtained in [1], [6], [12], [17], [19], [20], [21], [22], [23], [24], [25], [26].

In [14], Hong and Meng defined another index to measure the reliability of networks.

Definition 1.1 [14]

A graph G is said to be m-super edge connected (m-super-λ for short) if G − S is super-λ for any S ⊆ E(G) with ∣S∣ ⩽ m.

From the definition, we know that G is 0-super-λ is equivalent to that G is super-λ. Furthermore, if G is a-super-λ, then G is also b-super-λ, for any 0 ⩽ b ⩽ a. So m-super-λ is a generalization of super-λ.

The edge fault tolerance of super edge connectivity of G is an integer m such that G is m-super-λ but not (m + 1)-super-λ, denoted by S_λ(G).

In [14], Hong and Meng gave an upper and lower bound for S_λ(G). Moreover, more refined bounds for S_λ(G) of Cartesian product graphs, edge transitive graphs and regular graphs are given.

In this paper, we will give some bounds of S_λ(G) for three families of interconnection networks.

Before proceeding, we introduce some notions which will be used in the discussions in the next sections. Let G = (V, E) be a graph. For two disjoint vertex sets U₁, U₂ ⊆ V(G), we use [U₁, U₂]_G to denote the edge set of G with one end in U₁ and the other end in U₂. For any vertex set A ⊆ V(G), denote $ω_{G} (A) = |{[A, \bar{A}]}_{G}|$ , where $\bar{A} = V (G) - A$ is the complement of A. The subscription G is omitted when the graph under consideration is obvious. Next we cite two lemmas which will be used in the following proofs.

Lemma 1.2 [14]

A graph G is super-λ if and only if ω(A) > δ(G) for any A ⊂ V(G) with $2 ⩽ | A | ⩽ ⌊\frac{| V (G) |}{2}⌋$ and G[A] and $G [\bar{A}]$ being connected.

Lemma 1.3 [14]

Let G be a connected graph with minimum degree δ(G). Then S_λ(G) ⩽ δ(G) − 1.

Section snippets

Three families of interconnection networks

The following three families of interconnection networks which we will discuss in the next sections were introduced in [4].

The first family G(G₀, G₁; M) of networks

In this section, we will give lower bound of S_λ(G) for G = G(G₀, G₁; M).

Theorem 3.1

Let G_i = (V_i, E_i) be a connected graph of order n with δ_i = δ(G_i) = λ(G_i) = λ_i ⩾ 2, i = 0, 1. Let G = G(G₀, G₁; M) and $\bar{δ} = \min {δ_{0}, δ_{1}}$ . Then $S_{λ} (G) ⩾ \{\begin{matrix} \bar{δ} - 2 & if \bar{δ} ⩽ \frac{n}{2}, \\ n - \bar{δ} - 2 & if \frac{n}{2} ⩽ \bar{δ} ⩽ n - 2 . \end{matrix}$

Proof

Set $m = \min \{\bar{δ} - 2, n - \bar{δ} - 2\}$ . Then m ⩾ 0. We will show that G is m-super-λ, that is, for any S ⊆ E(G) with ∣S∣ ⩽ m, G − S is super-λ. Let S ⊆ E(G) with ∣S∣ ⩽ m, G′ = G − S and A a vertex set of V(G) with $2 ⩽ | A | ⩽ ⌊\frac{| V (G) |}{2}⌋ = n$ and G′[A] and $G^{'} [\bar{A}]$ being connected.

If G₀ and G₁ are disconnected in $G - [A, \bar{A}]$ ,

The second family $G (G_{0}, G_{1}, \dots, G_{r - 1}; \tilde{M})$ of networks

In this section, we will give the lower bound of S_λ(G) for $G = G (G_{0}, G_{1}, \dots, G_{r - 1}; \tilde{M})$ .

Theorem 4.1

Let G_i = (V_i, E_i) be a connected graph of order n with δ_i = δ(G_i) = λ(G_i) = λ_i ⩾ 2, i = 0, 1, … , r − 1 and r ⩾ 3. Let $G = G (G_{0}, G_{1}, \dots, G_{r - 1}; \tilde{M})$ and $\bar{δ} = \min {δ_{0}, δ_{1}, \dots, δ_{r - 1}}$ . Then $S_{λ} (G) ⩾ \bar{δ} - 1 .$

Proof

We will show that for any S ⊆ E(G) with $| S | ⩽ \bar{δ} - 1, G^{'} = G - S$ is super-λ.

Let A be a vertex set of V(G) with $2 ⩽ | A | ⩽ ⌊\frac{| V (G) |}{2}⌋$ and G′[A] and $G^{'} [\bar{A}]$ being connected. We will complete the proof by considering the following three cases.

Case 1.
G_i is connected in $G - [A, \bar{A}]$ for 0 ⩽ i ⩽ r − 1.
In this

The third family SP_n of networks

Next we consider the value of S_λ(G) for G = SP_n.

Lemma 5.1

G is an (n − 1)-regular graph with ∣V(G)∣ = n! and λ = n − 1.

Proof

By the definition of G, G is (n − 1)-regular and ∣V(G)∣ = n! We will show, by induction on n, that λ = n − 1. If n = 3 or n = 4, then it is easy to check that λ = δ(G) = n − 1 (see Fig. 3(a) and (b)).

Let n ⩾ 5. Note that λ ⩽ δ(G) = n − 1. To show that λ ⩾ n − 1, we just need to prove that for any edge set S ⊆ E(G) with ∣S∣ ⩽ n − 2, G − S is connected. By the definition of SP_n, for any i ≠ j (1 ⩽ i, j ⩽ n), there is an edge set $E_{i, j} = [V ({SP}_{n - 1}^{i}), V]$

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (Nos. 10971114 and 10990011). The authors are thankful to anonymous referees for their useful comments.

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Edge fault tolerance of super edge connectivity for three families of interconnection networks

Abstract

Introduction

Section snippets

Three families of interconnection networks

The first family G(G0, G1; M) of networks

The second family GG0,G1,…,Gr-1;M∼ of networks

The third family SPn of networks

Acknowledgments

Discrete Appl. Math.

Appl. Math. Comput.

Appl. Math. Comput.

Inform. Sci.

Inform. Process. Lett.

Discrete Math.

Discrete Math.

Discrete Math.

Inform. Sci.

Inform. Process. Lett.

Appl. Math. Comput.

Discrete Math.

Inform. Sci.

The first family G(G₀, G₁; M) of networks

The second family $G (G_{0}, G_{1}, \dots, G_{r - 1}; \tilde{M})$ of networks

The third family SP_n of networks