Edge fault tolerance of super edge connectivity for 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 λk(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 λk(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 Sλ(G).
In [14], Hong and Meng gave an upper and lower bound for Sλ(G). Moreover, more refined bounds for Sλ(G) of Cartesian product graphs, edge transitive graphs and regular graphs are given.
In this paper, we will give some bounds of Sλ(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 , where 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 and G[A] and being connected. Lemma 1.3 [14] Let G be a connected graph with minimum degree δ(G). Then Sλ(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 Sλ(G) for G = G(G0, G1; M). Theorem 3.1 Let Gi = (Vi, Ei) be a connected graph of order n with δi = δ(Gi) = λ(Gi) = λi ⩾ 2, i = 0, 1. Let G = G(G0, G1; M) and . Then Proof Set . 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 and G′[A] and being connected. If G0 and G1 are disconnected in ,
The second family of networks
In this section, we will give the lower bound of Sλ(G) for . Theorem 4.1 Let Gi = (Vi, Ei) be a connected graph of order n with δi = δ(Gi) = λ(Gi) = λi ⩾ 2, i = 0, 1, … , r − 1 and r ⩾ 3. Let and . Then Proof We will show that for any S ⊆ E(G) with is super-λ. Let A be a vertex set of V(G) with and G′[A] and being connected. We will complete the proof by considering the following three cases. Gi is connected in for 0 ⩽ i ⩽ r − 1. In this
The third family SPn of networks
Next we consider the value of Sλ(G) for G = SPn. 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 SPn, for any i ≠ j (1 ⩽ i, j ⩽ n), there is an edge set
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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2017, Theoretical Computer ScienceCitation Excerpt :In Section 2, we first introduce the notion of conditional arc connectivity as an extension of restricted edge connectivity to directed graphs, and then give a sharp upper bound on conditional arc connectivity. The other parameter for measuring super-λ property is the super-λ tolerance to edge-faults, which was first introduced by Hong et al. [6] and subsequently studied by several authors [7,18]. By replacing edges by arcs, one may introduce the concept of super-λ tolerance to arc-faults.