Strong Menger Connectivity on the Class of Hypercube-like Networks.

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(1)Strong Menger Connectivity on the Class of Hypercube-like Networks. Lun-Min Shih, Jimmy J.M. Tan Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 30050, R.O.C. lmshih@cs.nctu.edu.tw Abstract Motivated by some research works in networks with faults, we are interested in the connectivity issue of a network. Suppose that a network G has a set F of faulty vertices and let G−F is the resulting network with those vertices in F removed. We say that a k-regular graph G is strongly Menger connected if each pair u and v of G−F are connected by min{degf (u), degf (v)} vertex-disjoint fault-free paths in G−F, where degf (u) and degf (v) are the degree of u and v in G − F, respectively. In this paper, we consider an n-dimensional Hypercube-like networks HLn. We show that for each pair u and v in HLn−F, where F is a set of vertices, for |F| ≦ n−2, there are min{degf (u),degf (v)} vertex-disjoint fault- free paths connecting u to v, for n≧3.. 1: Introduction For the graph definitions and notations we follow [8]. A graph is denoted by G with the vertex set V(G) and the edge set E(G). A graph G is connected if there is a path between any two distinct nodes. A subset S of V(G) is a cut set if G − S is disconnected. The connectivity of G, written 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. A graph G is k-connected if k≦κ(G). We say that a graph has connectivity k if it is k-connected but not (k + 1)-connected. In this paper, we study the Hypercube-like networks, and show that it has a stronger connectivity property. Let G0 = (V0, E0) and G1 = (V1, E1) be two disjoint graphs with the same number of vertices. A one-to-one connection between G0 and G1 is defined as an edge set M = {(v,ψ(v)) | v ∈ V0, ψ(v)∈ V1 andψ: V0→V1 is a bijection}. We use G0 ♁M G1 to denote G=(V0∪V1, E0∪E1∪M). The. operation “ ♁ M” may generate different graphs depending on the bijection ψ. Chen et al. [2, 3] showed that the connectivity of G1 ♁ M G2 is increased to k+1, where both G1 and G2 have connectivity k. By the classical Menger’s Theorem [7], if a network G is k-connected, then every pair of vertices in G are connected by k vertex-disjoint paths. We can infer that there are k+1 vertex-disjoint paths between u and v, where u and v are vertices in G1 ♁M G2. The degree of a vertex u, denoted by deg(u), is the number of vertices adjacent to u. Suppose that the network G has a set F of faulty vertices and let G − F be the resulting network with those vertices in F removed. Let u and v be two fault-free vertices in G − F. The number of fault-free neighbor of u and v is denoted by degf (u) and degf (v), respectively. These values restrict the maximum number of vertex- disjoint fault-free path between u and v. It is clearly that the number of vertex-disjoint fault-free path between u and v is smaller than min{degf (u), degf (v)}. OH et al. [9] gave a definition to extend the Menger’s Theorem as following: Definition 1. [9] A k-regular graph G is strongly Menger-connected if for any copy G − F of G with at most k − 2 vertices removed, each pair u and v of G − F are connected by min{degf (u), degf (v)} vertex-disjoint fault-free paths in G − F, where degf (u) and degf (v) are the degree of u and v in G − F, respectively. This property is called strongly Menger connected property. OH et al. [9, 10] showed that the Star graph Sn with at most n − 3 vertices removed still possess the strongly Menger connected property. In this paper, we are interested in the strict bound on the size of the faulty vertex set F such that there are min{degf (u), degf (v)} vertex-disjoint fault-free paths connecting u to v, for each pair u and v in G − F. There are many useful topologies proposed in interconnection networks. Among them, the. - 546 -.

(2) hypercube network is one of the popular ones. Various networks are proposed by twisting some pairs of links in hypercubes [1, 6, 4, 5]. To make a unified study of these variants, Vaidya et al. [14] offered a class of graphs, called a class of hypercube-like graphs. The class of Hypercube-like networks consists of simple, connected and undirected graphs, and contains most of the hypercube variants. Park et al. [12, 13] showed some properties of Hypercube-Like Networks. We now give a recursive definition of the n-dimensional Hypercube-like networks HLn as follows: (1)HL0 = {k1}, where k1 is a trivial graph in the sense that it has only one vertex. (2)G ∈ HLn if and only if G = G0 ♁M G1 for some G0, G1 ∈ HLn−1 By the above definitions if G is a graph in HLn, then G = G0 ♁M G1 with both G0 and G1 in HLn−1, for n≧1. Let u be a vertex in V(G0). The vertex u has only one neighbor in V(G1). The connectivity of an n-dimensional Hypercube-like networks HLn is n. In this paper, we shall show that there are min{degf (u), degf (v)} vertex-disjoint fault-free paths connecting u to v, for each pair vertices u and v in HLn − F, where F is a set of vertices with |F|≦ n−2. This result is optimal in the sense that the result can not be guaranteed, if there are n−1 faulty vertices. For example, take an edge (x, y) and a vertex z different from x and y in HLn. Suppose that all the (n−1) vertices adjacent to x are faulty. Then degf (y) = degf (z) = n, (See Fig. 1). However, the number of vertex-disjoint paths between y and z is at most n−1, and hence there do not exist n vertex-disjoint paths connecting y and z.. Figure 1: four different output states.. 2: Some Preliminaries To prove our main theorem, we need the following fact. Lemma 1. Let HLn be an n-dimensional Hypercube-like networks with n≧3, and T be a set of vertices of HLn such that |T| ≦ 2n−3. Then HLn − T satisfies either (1) HLn − T is connected or (2) HLn − T has two components, one of which is a trivial graph.. Proof. We prove this statement by induction on n. We check this result for n=3 by brute force. Assume the lemma holds for n−1, for some n≧4, we shall show that it is true for n. As we mentioned before, we assume that G =G0 ♁ M G1 in HLn. So Gi ∈ HLn−1 for i = 0, 1. The connectivity of an n-dimensional Hypercube-like networks HLn is n, and HLn−1 has connectivity n−1. Thus, both G0 and G1 are (n−1)-connected. Let T0 and T1 be a set of faulty vertices of G0 and G1, respectively. By assumption, |T0| + |T1| = |T| ≦ 2n−3. The proof is divided into three major cases: Case 1: |T0| ≦ n−2 and |T1| ≦ n−2. Since G0 and G1 are both (n−1)-connected, then G0 − T0 and G1 − T1 are connected. There are 2n−1 edges between G0 and G1. For n≧4, since 2n−1−2(n − 2) ≧ 1, there is at least one edge with both ends fault-free remaining between G0 − T0 and G1 − T1. Hence HLn − T is also connected. Case 2: n−1≦|T0|≦2n−5 or n−1≦|T1|≦2n−5. Without loss of generality, we assume n−1≦|T0|≦ 2n−5, then |T1| ≦ n−2. So G1 − T1 is connected. We then consider that G0 − T0 is either connected or has two components, one of which has exactly one vertex. Assume first that G0 − T0 is connected. Then by the same reason of Case 1, G0 − T0 is connected to G1 − T1, since 2n−1 − (2n−3) ≧ 1 with n ≧4. Thus, HL n− T is also connected. On the other hand, if G0 − T0 is not connected, by the induction hypothesis, G0 − T0 has two components, C1 and C2, with C1 having only one vertex, we denote the vertex by x. For x≧4, since 2n−1 − 1 − (2n−3) ≧1, it means that there is at least one edge between C2 and G1 − T1. Hence C2 is connected to G1 − T1. If x (component C1) is connected to G1 − T1, then HLn − T is also connected. Otherwise, x is not connected to G1 − T1, then HLn − T is disconnected. We conclude that it has two components, one of which is trivial graph. Case 3: |T0|≧2n−4 or |T1|≧2n−4. Without loss of generality, we assume |T0|≧ 2n − 4. Since |T| ≦ 2n−3, we have either |T0| = 2n−4 or |T0| = 2n−3. If |T0| = 2n − 3, since every vertex of G0 has a neighbor in G1, then HLn − T is connected. Otherwise, we consider the last case |T0| = 2n−4. Since |T| = 2n−3 and |T0| + |T1| = |T|, so |T1| =1 and there is only one faulty vertex in G1, denoted by s. Let C be a connected component in G0 − T0. If C has at least two vertices, then C has at least two neighbors in G1. Since s is the only one vertex in G1, it infers that C is connected to G1 − T1. If C has only one vertex, denoted by t, then C is connected to G1 − T1 unless s is a neighbor of t. By the definition. - 547 -.

(3) of Hypercube-like networks, s has at most one neighbor in G0, so HLn − T is connected or HLn − T has exactly two components, one of which has exactly one vertex. □. 3 Main Theorem Our main results are presented in this section. Before proving the main theorem, we state the Menger Theorem. Theorem 1. [7] If x,y are vertices of a graph G and (x, y) ∉ E(G), then the minimum size of an x,y-cut equals the maximum number of pairwise internally disjoint x,y-paths. We now show that a Hypercube-like networks has a stronger connectivity property, it is strongly Menger-connected. Theorem 2. Consider an n-dimensional Hypercube-like networks HLn, for n≧3. Let F be a set of faulty vertices with |F| ≦ n−2. Then each pair vertices u and v of HLn − F are connected by min{degf (u), degf (v)} vertex-disjoint fault-free paths, where degf (u) and degf (v) are the degree of u and v in HLn − F, respectively. Proof. We can assume without loss of generality that degf (u)≦degf (v), so min{degf (u), degf (v)} = degf (u). To prove that each pair vertices u and v of HLn − F are connected by degf (u) vertex-disjoint fault-free paths, we show that u is connected to v if the number of vertices deleted is smaller than degf (u) − 1 in HLn − F. Suppose on the contrary that u and v is separated by deleting a set of vertices Vf , where |Vf |≦degf (u)−1. Obviously, |degf (u)−1| ≦ |deg(u)−1| ≦ n−1. So |Vf | ≦ n−1. We sum the cardinality of these two sets F and Vf . Since |F| ≦ n−2 and |Vf | ≦ n−1, then |F| + |Vf | = |T| ≦ 2n−3. By Lemma 1, HLn − T is either connected or has two components, one of which is a trivial graph, for |T| ≦ 2n−3. If HLn − T has two component and one of which has only one vertex, the set Vf has to be the neighbor of u and |Vf | = degf (u), which is a contradiction. Thus, u is connected to v when the number of vertices deleted is smaller than degf (u)−1 in HLn − F. This completes the proof. □. References. [1] S. Abraham and K. Padmanabhan, The twisted cube topology for multiprocessors: a study in network asymmetry, Journal of Parallel and Distributed Computing, 13, 1991, pp. 104-110. [2] Y. C. Chen, J. M. Tan, L. H. Hsu, and S. S. Kao, Super-connectivity and super-edge-connectivity for Some interconnection networks, Applied Mathematics and Computation, 140, 2003, pp. 245-254. [3] Y. C. Chen, C. H. Tsai, L. H. Hsu, and J. M. Tan, Super fault-tolerant hamiltonian graphs, Applied Mathematics and Computation, 148, 2004, pp. 729-741. [4] P. Cull and S. M. Larson, The M¨obius cubes, IEEE Transactions on Computers, 44, 1995, pp. 647-659. [5] K. Efe, The crossed cube architecture for parallel computing, IEEE Transactions on Parallel and Distributed Systems, 3, 1992, pp. 513-524. [6] A. H. Esfahanian, L. M. Ni, B. E. Sagan, The twisted n-cube with application to multiprocessing, IEEE Trans. Comput. , 40, 1991, pp. 88-93. [7] K. Menger, Zur allgemeinen kurventheorie, Fund. Math. 10, 1927, pp. 95-115. [8] U. S. R. Murty, Graph Theory with Applications, Macmillan Press, London, 1976. [9] E. OH and J. Chen, On Strong Menger-Connectivity of Star Graphs, Discrete Applied Mathematics, 129, 2003, pp. 499-511. [10] E. OH and J. Chen, Strong Fault-tolerance: Parallel routing in Star networks with Faults, 4, 2003, pp. 113-126. [11] C. D. Park and K. Y. Chwa, Hamiltonian propertities on the class of hypercube-like networks, Information Processing Letters, 91, 2004, pp. 11-17. [12] J. H. Park, H. C. Kim, and H. S. Lim, Fault-Hamiltonicity of Hypercube-Like Interconnection Networks, IPDPS archeive, in Proceedings of the 19th IEEE International Parallel and Distributed Processing Symposium, Vol.1, pp. 60.1. [13] J. H. Park, H. C. Kim, and H. S. Lim, Many to Many Disjoint Path Covers in Hypercube-Like Interconnection Networks with Faulty Elements, IEEE Transactions on Parallel and Distributed System, Vol. 17, No. 3, March 2006, pp. 227-240. [14] A. S. Vaidya, P. S. N. Rao, S. R. Shankar, A class of hypercube-like networks, in: Proc. of the 5th Symp. on Parallel and Distributed Processing, IEEE Comput. Soc. Los Alamitos, CA, December 1993, pp. 800-803.. - 548 -.

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