Many-to-many disjoint path covers in hypercube-like interconnection networks with faulty elements

A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k distinct source-sink pairs in which each vertex of G is covered by a path. We deal with the graph G/sub 0/ /spl oplus/ G/sub 1/ obtained from connecting two graphs G/sub 0/ and G/sub 1/ with n vertices each by n pairwise nonadjacent edges joining vertices in G/sub 0/ and vertices in G/sub 1/. Many interconnection networks such as hypercube-like interconnection networks can be represented in the form of G/sub 0/ /spl oplus/ G/sub 1/ connecting two lower dimensional networks G/sub 0/ and G/sub 1/. In the presence of faulty vertices and/or edges, we investigate many-to-many disjoint path coverability of G/sub 0/ /spl oplus/ G/sub 1/ and (G/sub 0/ /spl oplus/ G/sub 1/) /spl oplus/ (G/sub 2/ /spl oplus/ G/sub 3/ ), provided some conditions on the Hamiltonicity and disjoint path coverability of each graph G/sub i/ are satisfied, 0 /spl les/ i /spl les/ 3. We apply our main results to recursive circulant G(2/sup m/, 4) and a subclass of hypercube-like interconnection networks, called restricted HL-graphs. The subclasses includes twisted cubes, crossed cubes, multiply twisted cubes, Mobius cubes, Mcubes, and generalized twisted cubes. We show that all these networks of degree m with f or less faulty elements have a many-to-many k-DPC joining any k distinct source-sink pairs for any k /spl ges/ 1 and f /spl ges/ 0 such that f+2k /spl les/ m - 1.

[1]  Ke Qiu,et al.  From Hall's Matching Theorem to Optimal Routing on Hypercubes , 1998, J. Comb. Theory, Ser. B.

[2]  P. S. Nagendra Rao,et al.  A class of hypercube-like networks , 1993, Proceedings of 1993 5th IEEE Symposium on Parallel and Distributed Processing.

[3]  Jung-Heum Park.  Unpaired Many-to-Many Disjoint Path Covers in Hypercube-Like Interconnection Networks , 2006 .

[4]  Kyung-Yong Chwa,et al.  Recursive circulants and their embeddings among hypercubes , 2000, Theor. Comput. Sci..

[5]  Chang-Hsiung Tsai.  Linear array and ring embeddings in conditional faulty hypercubes , 2004, Theor. Comput. Sci..

[6]  Yu-Chee Tseng,et al.  Fault-Tolerant Ring Embedding in a Star Graph with Both Link and Node Failures , 1997, IEEE Trans. Parallel Distributed Syst..

[7]  Kanad Ghose,et al.  The Mcube: a symmetrical cube based network with twisted links , 1995, Proceedings of 9th International Parallel Processing Symposium.

[8]  Mihalis Yannakakis,et al.  Node-and edge-deletion NP-complete problems , 1978, STOC.

[9]  Lionel M. Ni,et al.  The Twisted N-Cube with Application to Multiprocessing , 1991, IEEE Trans. Computers.

[10]  Shietung Peng,et al.  Cluster fault-tolerant routing in star graphs , 1994, Networks.

[11]  Hyeong-Seok Lim,et al.  Fault-Hamiltonicity of hypercube-like interconnection networks , 2005, 19th IEEE International Parallel and Distributed Processing Symposium.

[12]  Hyeong-Seok Lim,et al.  Many-to-Many Disjoint Path Covers in Hypercube-Like Interconnection Networks with Faulty Elements , 2006, IEEE Trans. Parallel Distributed Syst..

[13]  A. Abdullah.  On graph bipartization , 1992, [Proceedings] 1992 IEEE International Symposium on Circuits and Systems.

[14]  Lutz Volkmann,et al.  Hamiltonian paths, containing a given path or collection of arcs, in close to regular multipartite tournaments , 2004, Discret. Math..

[15]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[16]  Paul Cull,et al.  The Möbius Cubes , 1995, IEEE Trans. Computers.

[17]  Kemal Efe.  A Variation on the Hypercube with Lower Diameter , 1991, IEEE Trans. Computers.

[18]  Gen-Huey Chen,et al.  Longest Fault-Free Paths in Star Graphs with Edge Faults , 2001, IEEE Trans. Computers.

[19]  Chi-Chang Chen,et al.  Nearly Optimal One-to-Many Parallel Routing in Star Networks , 1997, IEEE Trans. Parallel Distributed Syst..

[20]  Hyeong-Ah Choi,et al.  Graph Bipartization and via Minimization , 1989, SIAM J. Discret. Math..

[21]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[22]  Jimmy J. M. Tan,et al.  Fault hamiltonicity and fault hamiltonian connectivity of the arrangement graphs , 2004, IEEE Transactions on Computers.

[23]  Jung-Heum Park.  One-to-Many Disjoint Path Covers in a Graph with Faulty Elements , 2004, COCOON.

[24]  Ivan Hal Sudborough,et al.  A topological property of hypercubes: node disjoint paths , 1990, Proceedings of the Second IEEE Symposium on Parallel and Distributed Processing 1990.

[25]  Jimmy J. M. Tan,et al.  The super-connected property of recursive circulant graphs , 2004, Inf. Process. Lett..

[26]  Peter A. J. Hilbers,et al.  The Twisted Cube , 1987, PARLE.

[27]  Jung-Heum Park,et al.  Longest paths and cycles in faulty star graphs , 2004, J. Parallel Distributed Comput..

[28]  A. J. Nijman,et al.  PARLE Parallel Architectures and Languages Europe , 1987, Lecture Notes in Computer Science.

[29]  Kemal Efe,et al.  The Crossed Cube Architecture for Parallel Computation , 1992, IEEE Trans. Parallel Distributed Syst..

[30]  Fouad B. Chedid.  On the Generalized Twisted Cube , 1995, Inf. Process. Lett..

[31]  Kyung-Yong Chwa,et al.  Hamiltonian properties on the class of hypercube-like networks , 2004, Inf. Process. Lett..

[32]  Hyeong-Seok Lim,et al.  Many-to-many Disjoint Path Covers in a Graph with Faulty Elements , 2004, ISAAC.

[33]  Pao-Lien Lai,et al.  On the Two-Equal-Disjoint Path Cover Problem of Crossed Cubes , 2006, JCIS.

[34]  Gen-Huey Chen,et al.  Fault-Free Hamiltonian Cycles in Faulty Arrangement Graphs , 1999, IEEE Trans. Parallel Distributed Syst..

[35]  Cheng-Kuan Lin,et al.  The super laceability of the hypercubes , 2004, Inf. Process. Lett..

[36]  Jimmy J. M. Tan,et al.  Hamiltonian Properties of Faulty Recursive Circulant Graphs , 2002, J. Interconnect. Networks.

[37]  P. C. Treleaven,et al.  PARLE Parallel Architectures and Languages Europe , 1987, Lecture Notes in Computer Science.

[38]  Abhijit Sengupta.  On Ring Embedding in Hypercubes with Faulty Nodes and Links , 1998, Inf. Process. Lett..

[39]  Jung-Heum Park,et al.  Panconnectivity and pancyclicity of hypercube-like interconnection networks with faulty elements , 2007, Theor. Comput. Sci..